# Prescribing the behavior of Weil-Petersson geodesics in the moduli space of
Riemann surfaces

Babak Modami  Department of Mathematics, Yale University, 10 Hillhouse Ave,
New Haven, CT, 06511<EMAIL_ADDRESS>
###### Abstract.

We study the Weil-Petersson (WP) geodesics with narrow end invariant and
develop techniques to control length-functions and twist parameters along them
and prescribe their itinerary in the moduli space of Riemann surfaces. This
class of geodesics is rich enough to provide for examples of closed WP
geodesics in the thin part of moduli space, as well as divergent WP geodesic
rays with minimal filling ending lamination.

As an intermediate step we prove that hierarchy resolution paths between
narrow pairs of partial markings or laminations are stable in the pants graph
of the surface.

###### 2010 Mathematics Subject Classification:

Primary 30F60, 32G15, 37D40

The author was partially supported by NSF grant DMS-1005973

###### Contents

  1. 1 Introduction
    1. 1.1 Outline of the paper
  2. 2 Curve complexes and hierarchical structures
    1. 2.1 The pants and marking graph
    2. 2.2 Hierarchies and their resolutions in the pants and marking graph
    3. 2.3 $\Sigma-$hulls and their projections
  3. 3 The Weil-Petersson metric and its synthetic properties
    1. 3.1 End invariant
  4. 4 Length-function control along uniformly bounded length Weil-Petersson geodesics
    1. 4.1 Limits of sequences of uniformly bounded length WP geodesic segments
    2. 4.2 Length-function versus twist parameter control
  5. 5 Stable hierarchy paths
    1. 5.1 Narrow pairs
    2. 5.2 Stability
    3. 5.3 Fellow traveling
  6. 6 Itinerary of a Weil-Petersson geodesic segment
    1. 6.1 Isolated annular subsurfaces
    2. 6.2 Length-function control
  7. 7 Laminations with prescribed subsurface coefficients
    1. 7.1 Scheme I
    2. 7.2 Scheme II
  8. 8 Weil-Petersson Geodesics
    1. 8.1 Weil-Petersson geodesic rays with prescribed itinerary
    2. 8.2 Divergent geodesic rays
    3. 8.3 Closed geodesics in the thin part
    4. 8.4 A recurrence condition

##  1\. Introduction

The Weil-Petersson (WP) metric on the moduli space of Riemann surfaces is an
incomplete Riemannian metric with negative sectional curvatures asymptotic to
both $0$ and $-\infty$ in the completion. The WP metric is a metric of higher
rank in the sense of Gromov (see [BF06]). The WP geodesic flow is not
uniformly hyperbolic. These features prevent applying most of the standard
techniques available in the study of the global geometry and dynamics of
complete metrics with negative curvatures bounded away from zero and uniformly
hyperbolic dynamical systems to the WP metric and its geodesic flow. For some
of these techniques see [PP10], [Ebe72], [KH95]. The main theme of this paper
and the pioneer work of Brock, Masur and Minsky in [BM08],[BMM10],[BMM11] is
to apply combinatorial techniques from surface theory to study the global
behavior of WP geodesics.

In [BMM10] the authors introduce a notion of ending lamination for WP geodesic
rays. They show that the ending lamination determines the strong asymptotic
class of a WP geodesic ray recurrent to a compact subset of moduli space. In
[BMM11] a more explicit connection between the combinatorics of the ending
laminations of a WP geodesic and its behavior was established. They provide a
necessary and sufficient combinatorial condition for a WP geodesic to stay in
the compact part of moduli space. In this paper we prove the following two
results about the behavior of WP geodesics in the moduli space of Riemann
surfaces:

######  Theorem 1.1.

(Closed geodesic in the thin part) Given any compact subset of moduli space
$\mathcal{K}$, there are infinitely many closed Weil-Petersson geodesics not
intersecting $\mathcal{K}$.

######  Theorem 1.2.

(Divergent geodesic) Starting from any point in the moduli space there are
uncountably many divergent WP geodesic rays with minimal filling ending
lamination.

The WP volume of moduli space is finite, so by the Poincaré recurrence theorem
almost every WP geodesic ray is recurrent to a compact subset of moduli space.
However the second theorem above exhibits the abundance of WP geodesic rays
divergent in the moduli space. These geodesics diverge in the moduli space by
getting closer and closer to a chain of completion strata.

Given a WP geodesic $g:(a,b)\to\operatorname{Teich}(S)$ denote the end
invariant associated to its forward trajectory by $\nu^{+}=\nu^{+}(g)$ and the
one associated to its backward trajectory by $\nu^{-}=\nu^{-}(g)$. Here
$\nu^{-}$ and $\nu^{+}$ are either lamination or (partial) marking.

To each subsurface $Z\subseteq S$ that is not a three holed sphere, there is
an associated subsurface coefficient denoted by

$d_{Z}(\nu^{-},\nu^{+})$

which is the distance in the curve complex of $Y$ between the projections of
$\nu^{-}$ and $\nu^{+}$, for more details see $\S$2.

Subsurface coefficients are an analogue of continued fraction expansions which
provide for a coding of geodesics on the modular surface which is the moduli
space of one hold tori $\mathcal{M}(S_{1,1})$, see for example [Ser85].
Conjecturally these coefficients provide for extensive information about the
behavior of Weil-Petersson geodesics in the moduli space. The following
conjecture was proposed in [BMM10]:

######  Conjecture 1.3.

(Short Curve) Let $g$ be a Weil-Petersson geodesic with end invariant
$(\nu^{-},\nu^{+})$. Then

  1. (1)

For every $\epsilon>0$ there is an $A>0$, such that if
$d_{Z}(\nu^{-},\nu^{+})>A$ then $\inf_{t}\ell_{\alpha}(g(t))\leq\epsilon$ for
every $\alpha\in\partial{Z}$.

  2. (2)

For every $A>0$ there is an $\epsilon>0$, such that if
$\inf_{t}\ell_{\alpha}(g(t))\leq\epsilon$ then there is a subsurface
$Z\subsetneq S$ such that $\alpha\in\partial{Z}$ and
$d_{Z}(\nu^{-},\nu^{+})>A$.

Furthermore, it would be very useful to have estimates on the length of the
time interval that a curve is short (has length less than a given $\epsilon$)
along a WP geodesic using subsurface coefficients of the end invariants.

The WP metric exhibits different features in the thick and thin parts of
Teichmüller space. For instance in the thick part the sectional curvatures are
all bounded away from both 0 and $-\infty$, while in the thin part the WP
metric is almost a product metric. Therefore, to answer questions about the
global geometry and dynamics of the WP metric often one needs to determine the
itinerary of geodesics. By this we mean the thin regions of the Teichmüller
space that a WP geodesic $g$ visits, the order $g$ visits the regions and the
time $g$ spends in each one of these regions.

After the work of Masur-Minsky [MM99],[MM00] and Rafi [Raf05],[Raf], the above
conjecture holds for Teichmüller geodesics and provides for a complete picture
of the itinerary of Teichmüller geodesics in the moduli space. An important
ingredient which is missing here is an explicit description of the Riemann
surface along WP geodesics.

The underlying machinery to obtain control on the behavior of WP geodesics is
the Masur-Minsky machinery of hierarchies and their resolutions in pants and
marking graphs introduced in [MM00]. Given a pair of partial markings or
laminations a hierarchy (resolution) paths $\rho:[m,n]\to P(S)$,
$[m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\}$, is a quasi-geodesic between
them in the pants graph with certain properties encoded in the pair and their
subsurface coefficients. For example corresponding to any subsurface $Z$ with
big enough subsurface coefficient, there is a subinterval of $[m,n]$ denoted
by $J_{Z}$ such that $\partial{Z}\subset\rho(i)$, for every $i\in J_{Z}$. In
Theorem 2.13 the list of properties is listed.

By a result of Brock (Theorem 3.3) the Bers pants decompositions along a WP
geodesic trace a quasi-geodesic in the pants graph of the surface. When such a
quasi-geodesic and a hierarchy path $\rho$ fellow travel each other (see
Definition 5.23) there is a correspondence between the parameters of the
hierarchy path and the parameters of the WP geodesic, which is roughly the
nearest point correspondence of fellow traveling paths. In this situation we
use the hierarchy machinery to determine the itinerary of WP geodesics. An
example is visiting the region where the curves in $\partial{Z}$ are short
over the time interval corresponding to $J_{Z}$.

An $\it{A-narrow}$ condition on the end invariant $(\nu^{-},\nu^{+})$ is a
constraint on the set of subsurfaces with big subsurface coefficient. More
precisely, the pair is $A-$narrow if every non-annular subsurface $Z\subseteq
S$ with

$d_{Z}(\nu^{-},\nu^{+})>A$

is a large subsurface i.e. is a connected subsurface with complement
consisting of only annuli and three holed spheres.

The $A-$narrow condition on end invariant implies uniform fellow traveling,
depending only on $A$, of a WP geodesic segment and a hierarchy path with end
points equal to the pair. Heuristically hierarchy paths with narrow end
invariant avoid quasi-flats in the pants graph corresponding to separating
multi-curves on a surface, and WP geodesics with narrow end invariant avoid
asymptotic quasi-flats in the WP metric corresponding to separating multi-
curves on a surface. In this paper we develop a control for length-functions
and twist parameters along geodesics with narrow end invarinat and show that
their itinerary mimic combinatorial properties of hierarchy paths. In order to
prove our main technical results we introduce the following notions:

Let $Z\subseteq S$ be a subsurface with $d_{Z}(\nu^{-},\nu^{+})$ sufficiently
big. We say that $Z$ has $(R,R^{\prime})-$bounded combinatorics over a
subinterval $[i_{1},i_{2}]\subset J_{Z}\subset[m,n]$ if for every non-annular
subsurface $Y\subsetneq Z$ the subsurface coefficient be bounded as

$d_{Y}(\rho(i_{1}),\rho(i_{2}))\leq R$

and for every annular subsurface $A(\gamma)$ with core curve $\gamma$ inside
$Z$ the subsurface coefficient be bounded as

$d_{\gamma}(\rho(i_{1}),\rho(i_{2}))\leq R^{\prime}$

This condition is a local version of the bounded combinatorics of the end
invariant in [BMM11]. There is shown that a geodesic with bounded
combinatorics end invariant stays in the thick part of moduli space. In the
direction of Conjecture 1.3 we prove:

######  Theorem 1.5.

(Short Curve)

Given $A,R,R^{\prime}>0$ and a sufficiently small $\epsilon>0$, there is
$\bar{w}=\bar{w}(A,R,R^{\prime},\epsilon)$ with the following property. Let
$g:[a,b]\to\operatorname{Teich}(S)$ be a WP geodesic segment with $A-$narrow
end invariant $(\nu^{-},\nu^{+})$. Let $\rho:[m,n]\to P(S)$ be a hierarchy
between $\nu^{-}$ and $\nu^{+}$. Assume that a large component domain of
$\rho$, $Z$ has $(R,R^{\prime})-$bounded combinatorics over an interval
$[m^{\prime},n^{\prime}]\subset J_{Z}$.

If $n^{\prime}-m^{\prime}\geq 2\bar{w}$, then for every $\alpha\in\partial{Z}$
we have

$\ell_{\alpha}(g(t))\leq\epsilon$

for every $t\in[a^{\prime},b^{\prime}]$, where $a^{\prime}$ and $b^{\prime}$
are the corresponding times to $m^{\prime}+\bar{w}$ and $n^{\prime}-\bar{w}$,
respectively.

Here $N$ is the parameter map from $[m,n]$ to $[a,b]$ which comes from the
fellow traveling of a the Bers curves along a WP geodesic with narrow end
invariant $(\nu^{-},\nu^{+})$ and a hierarchy path between
$(\nu^{-},\nu^{+})$, see $\S$5.3.

In $\S$6 we use bounded combinatorics intervals to isolate annular subsurface
along a hierarchy path. The twist parameter developed about an isolated
annular subsurface along a WP geodesic is comparable to the one along a fellow
traveling hierarchy path. This together with the length-function versus twist
parameter controls over uniformly bounded length WP geodesic segments we
develop in $\S$4 are the main technical tools in $\S$6 where we prove the
above theorem.

Using the control we develop on length-functions along WP geodesic segments
and by extracting limits of WP geodesic segments with narrow end invariant we
provide WP geodesics with prescribed itinerary in the moduli space (Theorem
8.5). Itineraries of these rays mimic the combinatorial properties of
hierarchy paths encoded in the end points and the associated subsurface
cefficients. Using these rays and our constructions of pairs of
laminations/markings with prescribed list of subsurface coefficients in $\S$7
we prove theorems 1.1 and 1.2 in $\S$8. Jeffery Brock has communicated us that
he also constructs divergent WP geodesics, [Bro]. Our constructions of pairs
of laminations/markings with prescribed list of subsurface coefficients in
$\S$7 is a kind of symbolic coding for laminations using subsurface
coefficients analogue of continued fraction expansions.

Fellow traveling property of hierarchy paths is a crucial part of the
combinatorial frame work to control length-functions along WP geodesics. In
$\S$5 we prove the following stability result for hierarchy paths in the pants
graph of a surface

######  Theorem 5.11.

(Stable hierarchy path) Given $A>0$ there is a function $d_{A}$ such that any
hierarchy path with $A-$narrow end points is $d_{A}-$stable in the pants
graph.

Here $d_{A}:\mathbb{R}^{\geq 1}\times\mathbb{R}^{\geq 0}\to\mathbb{R}^{\geq
0}$ is the quantifier of the stability, see Definition 5.1.

This theorem implies that the Bers pants along a WP geodesic and a hierarchy
path connecting the end invariants of the geodesic uniformly fellow travel
each other in the pants graph.

###  1.1. Outline of the paper

Section 2 is devoted to the background about curve complexes and some
important notions and results in the setting of pants and marking graphs. Here
we recall hierarchical structures on pants and marking graphs and their
resolutions introduced by Masur and Minsky in [MM00]. In Theorem 2.13 we list
the properties of hierarchy resolution paths. Moreover, we recall the
$\Sigma-$hulls and their projections from [BKMM12]. In Section 3 we give some
background about the WP metric and synthetic properties of WP geodesics.

In Section 4 we provide the proofs of refined version of some of the key
results in [Wol03] and [BMM11] which we need in this work. These are mainly
based on compactness arguments in the WP completion of Teichmüller space and
are consequences of Wolpert’s geodesic limit theorem. These results give us a
kind of control on development of Dehn twists versus change off length-
functions along uniformly bounded length WP geodesic segments.

In Section 5 we prove that hierarchy paths between narrow pairs are stable.
The proof will be through $\Sigma-$hulls and their stability properties.

In Section 6, we develop some new techniques to control length-functions and
twist parameters along WP geodesics fellow traveling hierarchy paths. In Lemma
6.6 we provide certain rough bounds for length of curves. In Theorem 6.1 using
convexity of length-functions and an inductive argument on the complexity of
subsurfaces we sharpen these bounds.

In Section 7 we provide pairs of marking or laminations with a prescribed list
of subsurface coefficients. This is a kind of symbolic coding for laminations
in terms of subsurface coefficients similar to continued fraction expansions.

In Section 8 we provide examples of WP geodesics with prescribed behaviors.
These results could be considered as a kind of symbolic coding for WP
geodesics. Here we use the control on length-functions from $\S$6 and infinite
stable hierarchy paths with a prescribed list of subsurface coefficients from
$\S$7.

######  Notation 1.4.

Given $K\geq 1$ and $C\geq 0$. Let $f,g:X\to\mathbb{R}$ be two functions on a
set $X$. Then $f\asymp_{K,C}g$ means that for every $x\in X$ we have

$\frac{1}{K}g(x)-C\leq f(x)\leq Kg(x)+C$

Acknowledgment: I am so grateful to my thesis advisor Yair Minsky for so many
invaluable discussions through which this work has evolved. I would like to
thank Jeffery Brock who generously contributed some of the ideas in Section 4.
I am grateful to Scott Wolpert for willingly answering my questions about the
Weil-Petersson metric.

##  2\. Curve complexes and hierarchical structures

Curve complex of a surface: Let $S=S_{g,b}$ be a connected, compact, oriented
surface with genus $g$ and $b$ boundary components. We define $\xi(S)=3g-3+b$
as the complexity of the surface.

The curve complex of $S$, denoted by $\mathcal{C}(S)$, serves to organize the
isotopy classes of essential, simple closed curves on $S$. Let $S$ be a
surface with $\xi(S)\geq 1$. To each isotopy class of essential simple closed
curves on $S$ (neither isotope to boundary nor a point) is associated a vertex
($0-$simplex) in $\mathcal{C}(S)$. When $\xi(S)>1$, an edge is associated to
disjoint pair of isotopy classes of curves. Similarly, a $k-$simplex is
associated to any $k+1$ pairwise disjoint isotopy classes of simple closed
curves. Here two isotopy classes are disjoint if there are curves in each of
them which are disjoint on $S$. We denote the $k-$skeleton of this complex by
$\mathcal{C}_{k}(S)$. When $\xi(S)=1$, $S$ is either a one holed torus or a
four holed sphere. Then $1-$simplices (edges) correspond to curves
intersecting respectively once and twice, which are the minimum possible
intersection number of curves on $S$, respectively.

We equip the curve complex with a distance by making each simplex Euclidean
with side lengths 1, and denote the distance by $d_{S}=d_{\mathcal{C}(S)}$. By
the main result of Masur-Minsky in [MM99] $\mathcal{C}(S)$ equipped with this
distance is a $\delta-$hyperbolic space in the sense of Gromov, where $\delta$
only depends on topological type of $S$.

An annular subsurface is an annulus $Y$ with essential core curve in $S$. The
purpose of defining complexes for annuli is to keep track of Dehn twisting
about their core curves. These complexes are quasi-isometric to $\mathbb{Z}$.
Let $Y$ be an annulus with core curve. Let
$\widetilde{Y}=\widetilde{S}/\langle\alpha\rangle$ be the annular cover of $S$
to which $Y$ lifts homeomorphically. There is a natural compactification of
$\widetilde{Y}$ to a closed annulus $\widehat{Y}$ which is the quotient by
$\langle\alpha\rangle$ of the compactification of the universal cover of $S$,
$\widetilde{S}=\mathbb{D}^{2}$ (the Poincaré disk) by the closed disk. A
vertex of $\mathcal{C}(Y)$ is associated to a path connecting the two boundary
components of $\widehat{Y}$ modulo isotopies that fix the endpoints (isotopy
classes of arcs relative to the boundary). An edge is associated to two
vertices which have representatives with disjoint interiors. Curve comlexes
can be made metr with a metric by assigning length $1$ to each edge. Let
$\alpha\in\mathcal{C}_{0}(S)$ be the core curve of $Y$. We write $Y=A(\alpha)$
and $\mathcal{C}(\alpha)=\mathcal{C}(Y)$.

######  Notation 2.1.

A curve on $S$ refers to the isotopy class of an essential simple closed curve
on $S$, that is a vertex of $\mathcal{C}_{0}(S)$. If there are representatives
of the isotopy classes of curves $\alpha$ and $\beta$ which are disjoint, then
$\alpha$ and $\beta$ are disjoint. Otherwise, $\alpha$ and $\beta$ overlap
each other which is denoted by $\alpha\pitchfork\beta$. A multi-curve consists
of the vertices of a simplex in $\mathcal{C}(S)$, whihc is a collection of
pair-wise disjoint curves. A curve system is a finite diameter subset of
$\mathcal{C}_{0}(S)$. Two multi-curves $\sigma$ and $\sigma^{\prime}$ are
disjoint if any pair of curves $\alpha\in\sigma$ and
$\alpha^{\prime}\in\sigma^{\prime}$ are disjoint, otherwise $\sigma$ and
$\sigma^{\prime}$ overlap.

Filling curve systems: We say that a curve system $\mu$ on $S$ (a finite
diameter subset of $\mathcal{C}_{0}(S)$) fills $S$ if $S\backslash\mu$
consists of only topological disk and annuli. Given a filling system $\mu$ any
$\gamma\in\mathcal{C}_{0}(S)$ overlaps $\mu$.

Let $\alpha,\beta\in\mathcal{C}_{0}(S)$. If $d_{S}(\alpha,\beta)\geq 3$ then
$\alpha$ and $\beta$ fill $S$. To see this suppose that $\alpha$ and $\beta$
do not fill $S$. Then there is a curve $\gamma$ disjoint from both $\alpha$
and $\beta$ which implies that $d_{S}(\alpha,\beta)\leq 2$.

Laminations: Let $S$ be a surface equipped with a complete hyperbolic metric.
A geodesic lamination on $S$ is a closed subset $\lambda$ of $S$ consisting of
disjoint, complete, simple geodesics. We denote the space of geodesic
laminations of a surface $S$ by $\mathcal{GL}(S)$. A geodesic lamination
$\lambda$ can be equipped with transversal measures: a transversal measure is
a measure on arcs transversal to $\lambda$ which is invariant under isotopies
of the surface $S$ relative to $\lambda$. A measure (geodesic) lamination
$\mathcal{L}=(\lambda,m)$ is the pair of a geodesic lamination $\lambda$,
called the support of $\mathcal{L}$ and $m$ a transverse measure of $\lambda$.
The space of measured geodesic laminations of $S$ equipped with the weak∗
topology is denoted by $\mathcal{ML}(S)$. For more detail see [CEG87] and
[Bon01].

$\mathbb{R}^{+}$ acts on $\mathcal{ML}(S)$ by rescaling the measure, that is
$s(\lambda,m)=(\lambda,sm)$. Then the quotient,
$\mathcal{PML}(S):=\mathcal{ML}(S)/\mathbb{R}$ is the space of projective of a
measured laminations, the projective class of a measured lamination
$\mathcal{L}$ is denoted by $[\mathcal{L}]$. We equip $\mathcal{PML}(S)$ with
the quotient of the weak∗ topology on $\mathcal{ML}(S)$.

A geodesic lamination $\lambda$ is filling $S$ if the connected components of
$S\backslash\lambda$ are only topological disks or annuli. $\lambda$ is
minimal if any geodesic ray in $\lambda$ is dense in $\lambda$. Denote by
$\mathcal{EL}(S)$ the set of projective classes of measured laminations
supported on minimal filling laminations. Equip $\mathcal{EL}(S)$ with the
topology induced from the topology of $\mathcal{PML}(S)$.

The Gromov boundary of a $\delta-$hyperbolic space has a standard topology,
see for example $\S$III.H.3 of [BH99]. Klarriech in [Kla] proves that

######  Theorem 2.2.

There is a homeomorphism $\Phi$ from the Gromov boundary of $\mathcal{C}(S)$
to $\mathcal{EL}(S)$.

Furthermore Klariech describes the relation between a sequence of curves going
off to the Gromov boundary of $\mathcal{C}(S)$ and the accumulation points of
the sequence in $\mathcal{PML}(S)$.

######  Theorem 2.3.

(Theorem 1.4 of [Kla]) If a sequence of vertices in $\mathcal{C}_{0}(S)$
converges to a point $\xi$ in the Gromov boundary of $\mathcal{C}(S)$ then
regarding each element of the sequence as a projective measured lamination
every accumulation point of the sequence in $\mathcal{PML}(S)$ is
topologically equivalent to $\Phi(\xi)$. This means that their supporting
geodesic laminations are topologically equivalent.

The following notions of subsurface projection and subsurface coefficient are
basic in the Masur-Minsky machinery of curve complexes and hierarchical
structures on the pants and marking graphs.

Subsurface projection: Given a non-annular subsurface $Y\subseteq S$ define
the subsurface projection

$\pi_{Y}:\mathcal{GL}(S)\to\mathcal{P(C}_{0}(Y))$

where $\mathcal{P(C}_{0}(Y))$ is the power set of $\mathcal{C}_{0}(S)$ as
follows: Given $\lambda\in\mathcal{GL}(S)$, let $\lambda\cap Y$ be the
collection of all essential curves and arcs of the intersection of $\lambda$
and $Y$ after identifying all the arcs and curves which are isotopic to each
other. Here the end points of arcs are allowed to move within the boundary of
$Y$. $\pi_{Y}(\lambda)$ consists of the boundary of a regular neighborhood of
any arc $a$ in $\lambda\cap Y$ and $\partial{Y}$ and all of the closed curves
in $\lambda\cap Y$. If $\lambda$ does not intersect $Y$ essentially then
$\pi_{Y}(\lambda)=\emptyset$.

Note that since $\mathcal{C}_{0}(S)\subset\mathcal{GL}(S)$, the above
projection in particular gives a projection

$\pi_{Y}:\mathcal{C}_{0}(S)\to\mathcal{P(C}_{0}(Y)).$

The projection for an annular subsurface $Y$ is defined as follows: If
$\lambda\in\mathcal{GL}(S)$ crosses the core of $Y$ transversely, then the
lift of $\lambda$ to $\widehat{Y}$ (the compactification of $\tilde{Y}$ the
annular cover of $Y$) has at least one component that connects the two
boundaries of $\widehat{Y}$. Then together these components make up a set of
diameter $1$ in $\mathcal{C}(Y)$. Let $\pi_{Y}(\lambda)$ be this set. If
$\lambda$ does not intersect $Y$ essentially (including the case that
$\lambda$ is the core of $Y$) then $\pi_{Y}(\gamma)=\emptyset$.

The projection of a curve system $\mu$ to a subsurface $Y$ is the union of
$\pi_{Y}(\alpha)$ for all $\alpha\in\mu$.

Subsurface coefficient: Let $\mu$ and $\mu^{\prime}$ be two laminations or
curves systems. Let $Y\subseteq S$ be a subsurface. We consider the following
notion of distance

(2.1)
$d_{Y}(\mu,\mu^{\prime}):=\min\\{d_{Y}(\gamma,\gamma^{\prime}):\gamma\in\pi_{Y}(\mu),\gamma^{\prime}\in\pi_{Y}(\mu^{\prime})\\}$

which provides for a useful notion of complexity of $\mu$ and $\mu^{\prime}$
from the point of view of the subsurface $Y$. We call it the Y subsurface
coefficient of $\mu$ and $\mu^{\prime}$.

For an annular subsurface $Y$ with core curve $\alpha$ we denote
$\pi_{\alpha}:=\pi_{Y}$ and define the annular coefficients of geodesic
laminations/curve systems $\mu$ and $\mu^{\prime}$ by (2.1) and denote it by
$d_{\alpha}$.

Subsurface coefficients play the role of continued fraction expansions for
coding of laminations (see 7).

The triangle inequality: Given a curve system $\mu\subset\mathcal{C}(Y)$ (a
finite diameter subsets of $\mathcal{C}_{0}(Y)$) denote by
$\operatorname{diam}_{Y}(\mu)$ the diameter of $\pi_{Y}(\mu)$ as a subset of
$\mathcal{C}(Y)$.

Let $\mu_{1},\mu_{2},\mu_{3}\subset\mathcal{C}(Y)$ be curve systems. Suppose
that $d_{Y}(\mu_{1},\mu_{2})=d_{Y}(\alpha,\beta)$ and
$d_{Y}(\mu_{2},\mu_{3})=d_{Y}(\beta^{\prime},\gamma)$, where
$\alpha\in\mu_{1}$, $\beta,\beta^{\prime}\in\mu_{2}$ and $\gamma\in\mu_{3}$.
Then by the triangle inequality we have that
$d_{Y}(\alpha,\beta)+d_{Y}(\beta,\beta^{\prime})+d_{Y}(\beta^{\prime},\gamma)\geq
d_{Y}(\alpha,\gamma)$. Now since
$d_{Y}(\beta,\beta^{\prime})\leq\operatorname{diam}_{Y}(\mu_{2})$ and
$d_{Y}(\alpha,\gamma)\geq d_{Y}(\mu_{1},\mu_{2})$ we conclude that

$d_{Y}(\mu_{1},\mu_{2})+d_{Y}(\mu_{2},\mu_{3})+\operatorname{diam}_{Y}(\mu_{2})\geq
d_{Y}(\mu_{1},\mu_{3}).$

We refer to this inequality as the triangle inequality.

Hausdorff convergence: Let $\lambda_{k}$ be a sequence of geodesic laminations
and $\xi$ be any accumulation point of $\lambda_{k}$’s in the Hausdorff
topology of closed subsets of $S$. The Hausdorff convergence and the
definition of subsurface coefficient imply that given subsurface $Z\subseteq
S$ and $\lambda^{\prime}\in\mathcal{GL}(S)$

$d_{Z}(\lambda^{\prime},\xi)\asymp_{1,1}d_{Z}(\lambda^{\prime},\lambda_{k}),$

for all $k$ sufficiently large.

######  Proposition 2.4.

Let $\mathcal{L}_{i}=(\lambda_{i},m_{i})$ be a sequence of measured
laminations. Suppose that $\mathcal{L}_{i}$ converge to
$\mathcal{L}=(\lambda,m)$ in the weak∗ topology. Let $\xi$ be the limit of a
subsequece $\lambda_{i}$’s in the Hausdorff topology of closed subset of the
surface. Then $\lambda\subseteq\xi$.

###### Proof.

Let $x\in\lambda$. Let $U$ be an open neighborhood of $x$. Let $a\subset U$ be
an arc transversal to $\lambda$. By weak∗ convergence
$\int_{a}\mathcal{L}_{i}\to\int_{a}\mathcal{L}$ as $i\to\infty$. Further
$\int_{a}\mathcal{L}>0$, which implies that for all $i$ sufficiently large
$\int_{a}\mathcal{L}_{i}>0$. Then $\lambda_{i}\cap a$, for otherwise
$\int_{a}\mathcal{L}_{i}=0$. So there is $x_{i}\in\lambda_{i}$ so that
$x_{i}\in U$. This implies that $x$ is in the Hausdorff limit of any
convergent subsequence of $\lambda_{i}$’s. ∎

######  Notation 2.5.

A subsurface refers to the isotopy class of a subsurface. If a subsurface $Y$
has a representative which is a subset of a representative of a surface $Z$,
then $Y$ is a subsurface of $Z$, we denote it by $Y\subseteq Z$. Given a
multi-curve $\sigma$, $\sigma\pitchfork Y$ means that at least one of the
curves in $\sigma$ is not isotopic to curve in the complement of $Y$. If
$\partial{Y}\pitchfork Z$ and $\partial{Z}\pitchfork Y$, we say $Y$ and $Z$
overlap, and denote it by $Y\pitchfork Z$.

###  2.1. The pants and marking graph

A pants decomposition $P$ on a surface $S$ is a maximal collection of pairwise
disjoint simple closed curves. A (partial) marking $\mu$ consists of a pants
decomposition which is the base of the marking and is denoted by
$\operatorname{base}(\mu)$ together with an element of $\mathcal{C}(\alpha)$
for (some) every $\alpha\in\operatorname{base}{\mu}$. The element in
$\mathcal{C}(\alpha)$ can be represented by a transversal curve
$\beta(\alpha)$ to $\alpha$ on $S$. A clean marking is a marking such that
each transversal curve $\beta(\alpha)$ ($\alpha$ is in the base) does not
intersect any curve in $P-\alpha$ and has minimal intersection number 1 or 2
with $\alpha$ depending on that the subsurface $S\backslash\\{P-\alpha\\}$ is
a once punctured torus or a four holed sphere.

Pants and marking graph: Each vertex of the pants graph is a pants
decomposition, each edge corresponds to two pants decompositions which differ
by an elementary move. Pants decomposition $P$ and $P^{\prime}$ differ by an
elementary move if $P^{\prime}$ is obtained from $P$ by replacing one curve
$\alpha\in P$ with a curve $\alpha^{\prime}\in\mathcal{C}_{0}(S\backslash
P-\alpha)$ with minimal intersection number (1 or 2) with $\alpha$ and fixing
the rest of curves in $P$. Assigning length $1$ to each edge defines the
distance $d$ on $P(S)$ and makes it a metric graph.

Each vertex of the marking graph is a marking, each edge corresponds to a pair
of markings which differ by an elementary move. An elementary move on a
marking roughly speaking is either an elementary move on the base of the
marking or is an interchange of a curve in the base and its transversal curve.
For more detail see [MM00]. Assigning length one to each edge defines the
distance $\tilde{d}$ and makes the marking graph a metric graph.

The following theorem plays an important role in organizing the so called
tight geodesics in curve complexes of a surface and its subsurfaces and
inductive construction of hierarchies in the pants and marking graphs of a
surface.

######  Theorem 2.6.

(Bounded Geodesic Image)[MM00] There exists a constant $G>0$ depending only on
the topological type of $S$ with the following property. Let $Y\subsetneq S$
be an essential, connected subsurface and let $g:I\to\mathcal{C}(S)$ be a
geodesic such that for every $i\in I$, $\pi_{Y}(g(i))\neq\emptyset$ i.e.
$g(i)\pitchfork Y$. Then we have

$\operatorname{diam}_{Y}(\\{g(i)\\}_{i\in I})\leq G$

In the rest of this subsection we recall some results in the context of pants
and marking graphs which we use often in this paper.

Distance formula: Let $A\geq 0$, define the cut-off function
$\\{\\}_{A}:\mathbb{R}\to\mathbb{R}^{\geq 0}$ by
$\\{a\\}_{A}=\Big{\\{}\begin{array}[]{cl}a&a\geq A\\\ 0&a<A\\\ \end{array}.$
Masur-Minsky in [MM00] proved the following quasi distance formula for the
pants and marking graph distance.

There exists a constant $M_{1}>0$ such that for any $A\geq M_{1}$, there are
$K\geq 1$ and $C\geq 0$ such that the distance between any two pants
decompositions $P$ and $Q$ is given by

(2.2) $d(P,Q)\asymp_{K,C}\sum_{\begin{subarray}{c}Y\subseteq S\\\
\operatorname{non-annular}\end{subarray}}\\{d_{Y}(P,Q)\\}_{A}.$

Note that the sum is only over non-annular subsurfaces. We call $A$ the
threshold constant and say that $K,C$ are the constants corresponding to the
threshold constant $A$.

######  Remark 2.7.

Given partial markings $\mu$ and $\mu^{\prime}$ if in (2.2) we sum over all
subsurfaces coefficients, including the annular subsurface coefficients, we
get the marking distance of $\mu$ and $\mu^{\prime}$.

######  Theorem 2.8.

(Behrstock Inequality)[Beh06]

There exists a positive constant $B_{0}$ with the property that for
subsurfaces $Y,Z\subseteq S$ such that $Y\pitchfork Z$ and a partial marking
$\mu$ such that $\mu\pitchfork Y$ and $\mu\pitchfork Z$ we have

$\min\\{d_{Y}(\partial{Z},\mu),d_{Z}(\partial{Y},\mu)\\}\leq B_{0}.$

We recall the Consistency Theorem of Behrstock and Minsky from [BKMM12]. Note
that there this theorem is stated and proved for markings. It is
straightforward to verify that all of their arguments go through for pants
decompositions, considering only non-annular subsurfaces (excluding all
annular subsurfaces).

######  Theorem 2.9.

(Consistency)([BKMM12] Theorem 4.3) Given $F_{1},F_{2}\geq 1$, there is a
constant $F>0$ with the following property. Let $(x_{Y})_{Y\subseteq S}$ be a
tuple where $x_{Y}\in\mathcal{C}(Y)$ for each non-annular subsurface
$Y\subseteq S$. Suppose that $(x_{Y})_{Y\subseteq S}$ satisfies the following
two conditions:

  1. 1)

If $Y\pitchfork Z$ then
$\min\\{d_{Y}(x_{Y},\partial{Z}),d_{Z}(x_{Z},\partial{Y})\\}\leq F_{1}$, and

  2. 2)

If $Y\subseteq Z$ then $d_{Y}(x_{Y},\pi_{Y}(x_{Z}))\leq F_{2}.$

Then there is a $P\in P(S)$ such that for every subsurface $Y\subseteq S$ we
have:

$d_{Y}(P,x_{Y})\leq F.$

######  Remark 2.10.

Note that given a partial marking $\mu$, the tuple of $x_{Y}=\pi_{Y}(\mu)$
satisfies the above two conditions with $F_{1}=B_{0}$ ($B_{0}$ is the constant
from Theorem 2.8) and $F_{2}=1$.

We recall the following relation on subsurfaces of $S$ from [BKMM12].

######  Definition 2.11.

(Partial order in pants and marking graphs) Fix a tuple $(x_{Y})_{Y\subseteq
S}$ where $x_{Y}\subset\mathcal{C}(S)$ satisfying the consistency conditions
in Theorem 2.9. Assume that $F_{1}>\min\\{F_{2},B_{0},G\\}$, where $B_{0}$ is
the constant from Theorem 2.8. We define the following two relations on
proper, connected, essential subsurfaces of $S$:

  1. (i)

Given a positive integer $k$, $Y\prec_{k}Z$, if $Y\pitchfork Z$ and
$d_{Y}(x_{Y},\partial{Z})\geq k(F_{1}+4)$.

  2. (ii)

Given a positive integer $k$, $Y\ll_{k}Z$, if $Y\pitchfork\partial{Z}$ and
$d_{Y}(x_{Y},\partial{Z})\geq k(F_{1}+4)$

$Y\pitchfork Z$ if $\partial{Y}\pitchfork Z$ and $\partial{Z}\pitchfork Y$, so
it is immediate from the definition that $Y\prec_{k}Z\Longrightarrow
Y\ll_{k}Z$, but not the other way round. Moreover, if $k>l$ then
$Y\prec_{k}Z\Longrightarrow Y\prec_{l}Z$, and similarly
$Y\ll_{k}Z\Longrightarrow Y\ll_{l}Z$. These notions of partial order are
closely related to the partial order of subsurface in Proposition 2.15. The
following theorem provides for some useful transitivity properties of the
partial order defined above.

######  Theorem 2.12.

([BKMM12] Lemma 4.1)

Given an integer $k>1$, let $(x_{Y})_{Y\subseteq S}$ be a tuple satisfying the
consistency conditions and consider the partial order defined by it.

Let $U,V$ and $W$ be subsurfaces such that $x_{U},x_{V},x_{W}\neq\emptyset$
then we have:

  1. (1)

If $U\prec_{k}V$ and $V\ll_{2}W$ then $U\prec_{k-1}W$, also if $\mu$ is a
partial marking, and if $U\prec_{k}V$ and $V\ll_{2}\mu$, then
$U\prec_{k-1}\mu$.

  2. (2)

If $U\ll_{k}V$ and $V\ll_{2}W$ then $U\ll_{k-1}W$, also if $\mu$ is a partial
marking, and if $U\ll_{k}V$ and $V\ll_{2}\mu$ then $U\ll_{k-1}\mu$.

  3. (3)

If $U\pitchfork V$ and both $U\ll_{k}\mu$ and $V\ll_{k}\mu$ for a partial
marking $\mu$, then $U$ and $V$ are $\prec_{k-1}-$ordered, that is, either
$U\prec_{k-1}V$ or $V\prec_{k-1}U$.

###  2.2. Hierarchies and their resolutions in the pants and marking graph

The hierarchies of tight geodesics of curve complexes of subsurfaces of a
surface were introduced by Masur and Minsky in [MM00]. See also [Min10],
[BKMM12]. Hierarchy paths are quasi-geodesics in the pants and marking graph
of a surface, with quantifiers depending only on the topological type of $S$,
obtained by resolving hierarchies. Although in most of this paper we use only
hierarchy resolution paths and their properties given in Theorem 2.13, a good
understanding of the structure of hierarchies themselves will be extremely
useful for the reader to follow our arguments. Given a geodesic $h$ in the
curve complex of a subsurface $Y$, we refer to $Y$ as the support of $h$ and
denote it by $D(h)$. Let $(\mu^{-},\mu^{+})$ be a pair of partial makings or
laminations in $S$, a complete Hierarchy of tight geodesics $H$ inductively
associates to the pair $(\mu^{-},\mu^{+})$ a collections of tight geodesics in
the curve complex of $S$ and the curve complexes of its subsurfaces. A tight
geodesic in $\mathcal{C}(Y)$, is a geodesic with the property that any three
of its consecutive vertices fill the surface $Y$. Here we list the properties
which describe the inductive and layered structure of the hierarchy $H$, for
more detail see the above references.

  1. (1)

There is a unique main geodesic $g_{S}$ with $D(g_{S})=S$, whose endpoints lie
on $\operatorname{base}(\mu^{-})$ and $\operatorname{base}(\mu^{+})$.

  2. (2)

For any geodesic $h\in H$ other than $g_{S}$, there exists another geodesic
$k\in H$ such that, for some simplex $v\in k$, $D(h)$ is either a component of
$D(k)\backslash v$, or an annulus whose core is a component of $v$. We say
that $D(h)$ is a component domain of $k$.

  3. (3)

A subsurface in $S$ can occur as the domain of at most one geodesic in $H$.

Infinite hierarchies: An infinite hierarchy is a hierarchy $H$ with associated
infinite tight geodesic rays or lines in the curve complexes of subsurfaces.
Here each of the end points $\mu^{\pm}$ is a union of minimal, filling
laminations supported on disjoint subsurfaces. Each lamination is the point at
infinity of an infinite geodesic of the hierarchy in the curve complex of the
subsurface supporting the geodesic (see Theorem 2.2). The existence of
infinite hierarchies is proved in [Min10].

Hierarchy resolution paths: These are path which comprise for a set of
transitive quasi-geodesics in the pants (marking) graphs of the surfaces $S$
with constants depending only on the topological type of the surface. In $\S$5
we prove a condition for stability of hierarchy resolution paths in the pants
graph.

A pants graph resolution of a hierarchy $H$ is a path in the pants graph
denoted by $\rho:[n,m]\to P(S)$, where
$[m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\}$. For any $i\in[m,n]$,
$\rho(i+1)$ is obtained from $\rho(i)$ by an elementary move ($\rho(i+1)$ and
$\rho(i)$have pants distance $1$). Similarly a marking graph resolution of $H$
is a path of clean markings such that any two consecutive markings differ by
an elementary move. Given a hierarchy path $\rho$ we denote

$|\rho|=\\{\rho(i):i\in[n,m]\\}$

which is a subset of the pants graph. A resolution path of a hierarchy
consists of slices of the hierarchy. Each slice is the union of vertices of
geodesics in curve complexes of subsurfaces of $S$. The supporting domains of
these geodesics consist a tower of nested domains of $H$. In the following
theorem we some of the list properties of these quasi-geodesics (see also
[BMM11], [BM08]) which we will use frequently in this paper. The main feature
of these properties is that they are encoded in the subsurface coefficients of
the pair.

######  Theorem 2.13.

(Properties of pants hierarchy resolution paths)

Given partial markings or laminations $(\mu^{-},\mu^{+})$, there are hierarchy
(resolution) paths $\rho:[m,n]\to P(S)$
$([m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\})$ with $\rho(m)=\mu^{-}$ and
$\rho(n)=\mu^{+}$ satisfying the following list of properties. In what follows
$M_{1},M_{2}>0$ $(M_{1}\geq 2M_{2})$ are constants depending only on the
topological type of $S$.

  1. (1)

Given a component domain $Y$, there is a connected interval
$J_{Y}\subseteq[m,n]$ and a geodesic $g_{Y}\subset\mathcal{C}(Y)$ such that
for each $j\in J_{Y}$, $\partial{Y}\subset\rho(i)$ and there is a simplex
$v\in g_{Y}$ such that $v\in\rho(j)$ $(v\subset g_{Y}\cap\rho(j))$.

  2. (2)

There is a constant $M_{1}>0$, depending only on the topological type of $S$,
such that an essential, non-annular subsurface $Y\subseteq S$ with
$d_{Y}(\mu^{-},\mu^{+})>M_{1}$ is a component domain of $\rho$.

  3. (3)

(Monotonicity) Let $i,j\in J_{Y}$, $v=\rho(i)\cap g_{Y}$ and $w=\rho(j)\cap
g_{Y}$. Then $i\leq j$ if and only if $v\leq w$ as vertices along $g_{Y}$.

  4. (4)

(Bounded projection) Let $Y$ be a component domain of $\rho$ and
$J_{Y}=[j^{-},j^{+}]$. If $i>j^{+}$ then $d_{Y}(\rho(i),\rho(j^{+}))\leq
M_{2}$, and if $i<j^{-}$ then $d_{Y}(\rho(i),\rho(j^{-}))\leq M_{2}$.

  5. (5)

(Hausdorff distance bound) Let $W\subseteq S$ be a subsurface. For any
$i\in[m,n]$ there is an $x\in\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ such
that

$d_{W}(\rho(i),x)\leq M_{2}$

also for any $x\in\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ there is an
$i\in[m,n]$ such that the above bound holds. Here
$\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ is the convex hull of
$\pi_{W}(\mu^{-})$ and $\pi_{W}(\mu^{+})$ in $\mathcal{C}(W)$. In other words,
the Hausdorff distance of $\operatorname{hull}_{W}(\mu^{-},\mu^{+})$ and
$\pi_{W}(|\rho|)$ in $\mathcal{C}(W)$ is less than $M_{2}$. Note that here we
do not necessarily assume that $W$ is component domain of $\rho$. When $W$ is
a component domain the statement follows from (4).

  6. (6)

(No backtracking) Let $i,j,k\in[m,n]$ with $i\leq j\leq k$. Then for any
subsurface $Y\subseteq S$, $d_{Y}(\rho(i),\rho(k))+M_{2}\geq
d_{Y}(\rho(i),\rho(j))+d_{Y}(\rho(j),\rho(k))$.

In this theorem $J_{Y}\subset[m,n]$ consists of all $j\in[m,n]$ such that
$\rho(j)$ is a slice of $H(\mu^{-},\mu^{+})$ containing $(v,g_{Y})$, where
$g_{Y}\in H$ is the tight geodesic supported on $Y$ and $v\in g_{Y}$. The fact
that $J_{Y}$ is an interval was proved in Lemma 4.9 of [BCM12]. This explains
property (1). Property (2) is Lemma 6.1 (Large Link Lemma) in [MM00]. Property
(4) is a consequence of the Bounded Geodesic Image Theorem and is established
in the proof of Large Link Lemma. Property (5) is established in the proof of
Lemma 5.14 in [Min10]. Property (6) is a straight forward consequence of
properties (4) and (5). Property (3) is a consequence of the definition of
slices of hierarchy and the partial order of them defined using the partial
order of point geodesics of $H$ in $\S 5$ of [MM00].

######  Remark 2.14.

Given $(\mu^{-},\mu^{+})$, a marking hierarchy resolution path
$\tilde{\rho}:[m,n]\to M(S)$ $([m,n]\subseteq\mathbb{Z}\cup\\{\pm\infty\\})$
between $\mu^{-}$ and $\mu^{+}$ satisfies the same list of properties as pants
hierarchy resolution paths, besides in properties (1)-(6) subsurfaces $Y$ and
$W$ can be annular subsurfaces as well.

Partial order on subsurfaces along hierarchies is introduced in [MM00]. It
roughly gives the order in which the intervals $J_{Y}$ appear along the
component domains of a resolution of the hierarchy. In this paper we mainly
need the following weaker version of it.

######  Proposition 2.15.

(Order of subsurfaces) Fix the constant $M=M_{1}+B_{0}+4$. Let $Y,W\subseteq
S$ be subsurfaces. Suppose that $Y\pitchfork W$, $d_{Y}(\mu,\mu^{\prime})>4M$
and $d_{W}(\mu,\mu^{\prime})>4M$. Then one and only one of the following
inequalities holds.

  1. (a)

$d_{Y}(\mu,\partial{W})\geq 2M$ and $d_{W}(\mu^{\prime},\partial{Y})>2M$.

  2. (b)

$d_{W}(\mu,\partial{Y})\geq 2M$ and $d_{Y}(\mu^{\prime},\partial{W})>2M$.

If (a) holds we denote $Y<W$. If (b) holds we denote $W<Y$.

Furthermore, the relation $<$ has the following properties.

  * •

It is transitive.

  * •

Suppose that $W<Y$, $j\in J_{Y}-J_{W}$ and $i\in J_{W}-J_{Y}$, then $i\leq j$.

###### Proof.

Since $d_{Y}(\mu,\mu^{\prime})>4M$, by the triangle inequality either
$d_{Y}(\mu,\partial{W})\geq 2M$ or $d_{Y}(\mu^{\prime},\partial{W})\geq 2M$.
First assume that $d_{Y}(\mu,\partial{W})\geq 2M$ i.e. (a) holds. Then by
Theorem 2.8 (Behrstock Inequality) we get $d_{W}(\mu,\partial{Y})\leq
B_{0}<M$. So (b) can not hold. Furthermore, by the assumption of the
proposition, $d_{W}(\mu,\mu^{\prime})>4M$. The last two inequalities combined
by the triangle inequality imply that
$d_{W}(\mu^{\prime},\partial{Y})>3M-\operatorname{diam}_{W}(\mu)\geq 3M-2>2M$.
When $(b)$ holds a similar argument shows that $d_{Y}(\mu,\partial{W})\leq
B_{0}$, thus (a) does not hold and using the assumption
$d_{Y}(\mu,\mu^{\prime})>4M$ we get $d_{Y}(\mu^{\prime},\partial{W})>2M$.

To prove the transitivity of $<$, let $W<Y$ and $Y<Z$. Then as we saw above
$d_{Y}(\mu^{\prime},\partial{W})>2M$ and $d_{Z}(\mu^{\prime},\partial{Y})>2M$.
The second inequality and the Behrstock inequality imply that
$d_{Y}(\mu,\partial{Z})\leq M$. This and first inequality combined by the
triangle inequality imply that

$d_{Y}(\partial{W},\partial{Z})>M-2.$

But $M-2\geq 2$, thus $d_{Y}(\partial{W},\partial{Z})\geq 3$. So $\partial{Y}$
and $\partial{Z}$ fill $W$. Thus $\partial{Y}\pitchfork\partial{Z}$ and
$Y\pitchfork Z$. $M-2>B_{0}$ so by the above inequality we also have that
$d_{Y}(\partial{W},\partial{Z})>B_{0}$. Then by the Behrstock inequality
$d_{W}(\partial{Y},\partial{Z})\leq B_{0}$. Now we have

$\displaystyle d_{W}(\mu,\partial{Z})$ $\displaystyle\geq$ $\displaystyle
d_{W}(\mu,\mu^{\prime})-d_{W}(\partial{Y},\mu^{\prime})-d_{W}(\partial{Y},\partial{Z})-\operatorname{diam}_{W}(\partial{Y})-\operatorname{diam}_{W}(\mu^{\prime})$
$\displaystyle\geq$ $\displaystyle 4M-2B_{0}-3>2M.$

This finishes proving that $W<Z$.

We proceed to prove the last assertion of the proposition. Since $W<Y$,
$d_{W}(\partial{Y},\mu)\geq 2M$ so by Theorem 2.8, $d_{Y}(\partial{W},\mu)\leq
M$. Moreover, $i\in J_{W}-J_{Y}$ so $\rho(i)\supset\partial{W}$. Thus

$d_{Y}(\rho(i),\mu)\leq M(*).$

Now suppose that $i>j$, then since $j\in J_{Y}-J_{W}$, $j$ is greater than the
right end pint of the interval $J_{Y}$. So Theorem 2.13 (4) implies that
$d_{Y}(\rho(i),\mu^{\prime})\leq M$. This inequality combined by
$d_{Y}(\mu,\mu^{\prime})>4M$ with the triangle inequality implies that

$d_{Y}(\rho(i),\mu)>3M-2>M$

which contradicts the upper bound (*). Thus $i\leq j$ as was desired. ∎

In this paper we deduce almost all of the properties of hierarchy paths we
need from the Theorem 2.13 and Proposition 2.15. In a couple of occasions we
need some finer properties of hierarchies and their resolutions where we
provide a reference.

###  2.3. $\Sigma-$hulls and their projections

In this section we recall the subsets of pants graph called $\Sigma-$hull and
their projection introduced in [BKMM12]. The projection is coarsely the
closest point projection on the $\Sigma-$hull as a subset of the pants graph
of the surface. Note that in [BKMM12] these notions are introduced in the
context of marking graphs. Other places were variations of this projection is
used are [Beh06],[BM08],[BMM11]. Given a pair of partial markings or
laminations $(\mu^{-},\mu^{+})$ and $\epsilon>0$ define

$\Sigma_{\epsilon}(\mu^{-},\mu^{+}):=\\{P\in
P(S):d_{Y}(P,\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq\epsilon$ for every
non-annular subsurface $W\subseteq S\\}$

Here $\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ is the set of all geodesics in
$\mathcal{C}(Y)$ connecting $\pi_{Y}(\mu^{-})$ to $\pi_{Y}(\mu^{+})$. Note
that since $\mathcal{C}(Y)$ is $\delta_{Y}-$hyperbolic all of the geodesics
connecting $\pi_{Y}(\mu^{-})$ and $\pi_{Y}(\mu^{+})$ uniformly fellow travel
each other.

######  Theorem 2.16.

([BKMM12] Proposition 5.2) There is an $F>0$ depending only on the topological
type of the surface such that for every $\epsilon>F$ there is a coarse map
(projection)

$\Pi:P(S)\to\Sigma_{\epsilon}(\mu^{-},\mu^{+})$

with the following properties:

  1. (1)

For every non-annular subsurface $Y\subseteq S$ we have

$d_{Y}(\Pi P,\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq F$

  2. (2)

$\Pi\big{|}_{\Sigma_{\epsilon}(\mu^{-},\mu^{+})}$ is uniformly close to the
identity.

  3. (3)

$\Pi$ is coarse-Lipschitz.

######  Remark 2.17.

This theorem in [BKMM12] is stated and proved in the context of marking graph.
But it is straightforward to versify that all of their arguments go through in
the context of pants graph excluding all annular subsurfaces.

The coarse Lipschitz refers to the fact that the projection is defined on the
vertices of the pants graph and does not say any thing when two points are
within distance $1$.

Using the distance formula (2.2) $\Pi P$ coarsely minimizes the distance
between $P$ and the $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ in the pants graph.

The main ingredient of the proof of Theorem 2.16 is to prove that there are
positive constants $F_{1}$ and $F_{2}$, depending only on the topological type
of the surface, such that the tuple $(x_{Y})_{Y\subseteq S}$, where each
$x_{Y}\in\mathcal{C}(Y)$ is a nearest point to $\pi_{Y}(P)$ on
$\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$, satisfies the consistency
conditions of Theorem 2.9 (Consistency Theorem). Then the consistency theorem
implies that there is a constant $F>0$ and a pants decomposition, denoted by
$\Pi P$, such that $d_{Y}(\Pi P,\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq
F$ for every $Y\subseteq S$.

##  3\. The Weil-Petersson metric and its synthetic properties

We start with some basic facts about Teichmüller theory and the Weil-Petrsoon
(WP) metric, and through out will set up our notation. Let $S$ be a surface
with genus $g$ and $b$ boundary components. A point in the Teichmüller space
of $S$, denoted by $\operatorname{Teich}(S)$, is a complete, finite area
hyperbolic surface $x$ equipped with a diffeomorphism $h:S\to x$. The map $h$
is a marking for $x$. Two marked surfaces $h_{1}:S\to x_{1}$ and $h_{2}:S\to
x_{2}$ define the same point in $\operatorname{Teich}(S)$ if and only if
$h_{2}\circ h_{1}^{-1}:x_{1}\to x_{2}$ is isotopic to an isometry. The mapping
class group of the surface $S$, denoted by $\operatorname{Mod}(S)$, is the
group of isotopy classes of orientation preserving self diffeomorphisms of
$S$. $\operatorname{Mod}(S)$ acts on $\operatorname{Teich}(S)$ by remarking as
follows: an element $f\in\operatorname{Mod}(S)$ maps a marked surface $h:S\to
x$ to the marked surface $h\circ f:S\to x$. The quotient
$\operatorname{Teich}(S)/\operatorname{Mod}(S)$ is the moduli space of $S$
denoted by $\mathcal{M}(S)$. Given a point $f:S\to x$ in the Teichmüller space
we usually drop the marking map and denote it by $x$. We denote the point in
moduli space corresponding to the $\operatorname{Mod}(S)$ orbit of $x$ by
$\hat{x}$.

Given $\epsilon>0$, the $\epsilon-$thick part of Teichmüller space is
$\\{x\in\operatorname{Teich}(S):\operatorname{inj}(x)\geq\epsilon\\}$. Here
$\operatorname{inj}(x)$ is the injectivity radius of the hyperbolic surface
$x$. The $\epsilon-$thin part is
$\\{x\in\operatorname{Teich}(S):\operatorname{inj}(x)\leq\epsilon\\}$. Let
$\epsilon>0$ be small enough such that by the Collar lemma ($\S 4.1$ of
[Bus10]) on any complete hyperbolic surface there is not any pair of
intersecting closed geodesic with length less than or equal of $\epsilon$. Let
$\epsilon^{\prime}>0$. Given a multi-curve $\sigma$ (a simplex in
$\mathcal{C}(S)$) we define the following regions in the Teichmüller space:

  * •

$U_{\epsilon}(\sigma):=\\{x\in\operatorname{Teich}(S):\ell_{\alpha}(x)\leq\epsilon$
for any $\alpha\in\sigma\\}$

  * •

$U_{\epsilon,\epsilon^{\prime}}(\sigma):=\\{x\in\operatorname{Teich}(S):\ell_{\alpha}(x)\leq\epsilon$
for any $\alpha\in\sigma$ and $\ell_{\alpha^{\prime}}(x)>\epsilon^{\prime}$
for every $\alpha^{\prime}\notin\sigma\\}$

Here we recall some properties of the Weil-Petersson metric and its geodesics
which will be used in this paper. References for these material are [Wol03],
[Wol08],[Wol10], see them also for further references.

Given holomorphic quadratic differentials $\varphi,\psi\in
T_{x}^{*}\operatorname{Teich}(S)$ the Weil-Petersson $L^{2}$ co-product is
defined by

$\mathcal{R}e(\int_{x}\varphi\overline{\psi}\rho^{-2})$

where $\rho(z)^{2}|dz|^{2}$ is the hyperbolic metric of the marked hyperbolic
surface $x$. This co-product induces a norm on Teichmüller space via the
standard pairing of quadratic differentials and measurable Beltrami
differentials on $x$ whcih is defined by $\int_{x}\varphi\mu$. Any measurable
Beltrami differential presents a vector in $T_{x}\operatorname{Teich}(S)$ and
the Weil-Petersson metric on Teichmüller space is defined by the polarization
of this norm. In this paper we study the global behavior of geodesics of this
metric.

The Weil-Petersson metric is a Riemannian metric with negative sectional
curvatures which is invariant under the action of the mapping class group of
the surface. It is an incomplete metric, however it is geodesically convex.
The negative curvature and convexity imply that the completion of Teichmüller
space with respect to the WP metric $\overline{\operatorname{Teich}(S)}$ is a
$\operatorname{CAT}(0)$ space. For background about $\operatorname{CAT}(0)$
space see for example [BH99].

Length-functions: Given any $\alpha\in\mathcal{C}_{0}(S)$ the $\alpha-$length-
function

$\ell_{\alpha}:\operatorname{Teich}(S)\to\mathbb{R}^{+}$

assigns to $x\in\operatorname{Teich}(S)$ the length of the geodesic
representative of $\alpha$ on the marked hyperbolic surface $x$.

The notion of length-function has a natural extension to the space of measured
geodesic laminations (see [Bon01]). Given a measured geodesic lamination
$\mathcal{L}$ we denote the $\mathcal{L}-$length-function by
$\ell_{\mathcal{L}}(x)$.

Fenchel-Nielsen coordinates and twist parameters: Given a pants decomposition
$P$, a Fenchel-Nielsen (FN) coordinate system
$(\ell_{\gamma},\theta_{\gamma})_{\gamma\in P}$ maps $\operatorname{Teich}(S)$
to $\prod_{\gamma\in P}\mathbb{R}_{\gamma}^{+}\times\mathbb{R}_{\gamma}$. The
first coordinate of the pair
$\mathbb{R}_{\gamma}^{+}\times\mathbb{R}_{\gamma}$ is the $\gamma-$length-
function and the second coordinate is a twist parameter about $\gamma$. For
more detail about Fenchel-Nielsen coordinates and twit parameters see $\S 3$
of [Bus10]. We denote the Dehn twist about a curve $\gamma$ by
$\mathcal{D}_{\gamma}$, it is defined as follows: Let
$x\in\operatorname{Teich}(S)$. Given a pants decomposition $P$ with $\gamma\in
P$ fix a FN coordinate system, $\mathcal{D}_{\gamma}(x)$ is the point with all
coordinates equal to that of $x$ except
$\theta_{\gamma}(\mathcal{D}_{\gamma}(x))=\theta_{\gamma}(x)+2\pi$.

The Weil-Petersson completion of Teichmüller space and the completion strata:
The incompleteness of the Weil-Petersson metric is due to existance of finite
length paths in Teichmüller space along which length of a curve converges to
zero, [Wol10]. In [Mas76], Masur gives a concrete description of the
completion as the augmented Teichmüller space. The augmented Teichmüller space
consists of strata: Let $\sigma$ be a simplex in the augmented curve complex
$\widehat{\mathcal{C}(S)}=\mathcal{C}(S)\cup\emptyset$, a point in the
$\sigma-$stratum is a collection of marked hyperbolic metrics of connected
components of $S\backslash\sigma$, where for each curve in $\sigma$ a pair of
cusps is introduced. The topology is described via extended Fenchel-Nielsen
coordinate systems as follows: Given a pants decomposition $P$, the FN
coordinate system maps $\operatorname{Teich}(S)$ to $\prod_{\gamma\in
P}\mathbb{R}\times\mathbb{R}^{+}$. We extend the FN coordinate system
$(\ell_{\gamma},\theta_{\gamma})_{\gamma\in P}$ to allow length-functions take
value $0$ as well. Now take the quotient of $\prod_{\gamma\in
P}\mathbb{R}\times\mathbb{R}^{+}$ by identifying
$(0,\theta)\sim(0,\theta^{\prime})$ in each $\mathbb{R}^{+}\times\mathbb{R}$
factor. Let $\sigma\subset P$ then the topology near any point of the
$\sigma-$stratum is such that the map defined by the FN coordinate system is a
homeomorphism near that point.

In this topology each stratum $\mathcal{S}(\sigma)$ is the product of the
lower dimensional Teichmüller spaces of the connected components of
$S\backslash\sigma$.

Continuity of length-functions: In this paper we refer to the following
theorem as the continuity of length-functions. It is consequence of the fact
that the topology induced by the Weil-Petersson metric and the Chaubaty
topology of the Teichmüller space are the same. The Chaubaty topology is
defined using the fact that each point in Teichmüller space is the conjugacy
class of a representation of $\pi_{1}(S)$ into $PSL_{2}(\mathbb{R})$. For more
detail see the beginning of $\S 4$ of [Wol08].

######  Theorem 3.1.

(Continuity of length-functions) Suppose that a sequence of points $x_{n}\to
x$ as $n\to\infty$ in the the completion of Teichmüller space with respect to
the WP metric. Then for every $\alpha\in\mathcal{C}_{0}(S)$,
$\ell_{\alpha}(x_{n})\to\ell_{\alpha}(x)$ as $n\to\infty$.

Non-refraction of strata:

######  Theorem 3.2.

(Non-refraction)[DW03],[Wol03], [Yam04]

Let $\zeta:[0,T]\to\overline{\operatorname{Teich}(S)}$ be a WP geodesic
segment, and let $\sigma^{-}$ and $\sigma^{+}$ be the maximal simplicies in
$\mathcal{C}(S)$ such that $\zeta(0)\in\mathcal{S}(\sigma^{-})$ and
$\zeta(T)\in\mathcal{S}(\sigma^{+})$ then

$int(\zeta)\subset\mathcal{S}(\sigma^{-}\cap\sigma^{+})$

As a consequence of the Non-refraction Theorem Daskalopoulos and Wentworth in
[DW03] and Wolpert in [Wol03] show that any pseudo-Anosov element of the
mapping class group $f$ has an axis in the Teichmüller space equipped with the
WP metric. The axis is a bi-infinite WP geodesic
$Ax_{f}\subset\operatorname{Teich}(S)$ such that

$d_{\operatorname{WP}}(x,fx)=\inf_{y\in\operatorname{Teich}(S)}d_{\operatorname{WP}}(y,fy)$

for every $x\in\operatorname{Ax}_{f}$. The axis projects to a closed geodesic
in the moduli space.

Bers pants decomposition and Bers marking: By a result of Bers (see $\S 3$ of
[Bus10]) given a surface $S$ with $\chi(S)<0$, there is a constant $L_{S}>0$
(Bers constant) depending only on the topological type of $S$ such that any
complete finite area hyperbolic metric on $S$ posses a pants decomposition
(Bers pants decomposition) with the property that the geodesic representative
of any curve in the pants decomposition has length at most $L_{S}$. We call
any curve in a Bers pants decomposition a Bers curve. By the Collar Lemma
there are only finitely many Bers curves and consequently Bers pants
decompositions on a complete hyperbolic surface. Given
$x\in\overline{\operatorname{Teich}(S)}$, suppose that
$x\in\mathcal{S}(\sigma)$ ($\sigma$ is a simplex in
$\widehat{\mathcal{C}(S)}$). Then a Bers pants decomposition of $x$ is the
union of Bers pants decompositions of each of the connect components of
$S\backslash\sigma$ and $\sigma$. A Bers marking is a (partial) marking
obtained from a Bers pants decomposition by adding transversal curves with
representatives at $x$ of minimal length. We denote a Bers marking of
$x\in\overline{\operatorname{Teich}(S)}$ by $\mu(x)$. Given
$x\in\mathcal{S}(\sigma)$ the partial marking does not have any transversal to
the curves in $\sigma$.

By the following theorem of Brock the hierarchies of curves complexes and
their resolution in pants and marking graphs would play an essential role in
our study of the global behavior of WP geodesics:

######  Theorem 3.3.

(Quasi-isometric model)[Bro03] There are constants $K_{\operatorname{WP}}\geq
1$ and $C_{\operatorname{WP}}\geq 0$ depending only on the topological type of
$S$, such that the coarsely defined map

$Q:\overline{\operatorname{Teich}(S)}\to P(S)$

which assigns to $x$ a Bers pants decomposition $Q(x)$ is a
$(K_{\operatorname{WP}},C_{\operatorname{WP}})-$quasi-isometry.

Gradient of length-functions: Wolpert gives the following estimate for the
pairing of the gradients of length-functions:

######  Lemma 3.4.

[Wol08] The WP pairing of length-function gradients of curves $\alpha,\beta$
with disjoint geodesic representatives satisfies

$0<\langle\operatorname{grad}\ell_{\alpha},\operatorname{grad}\ell_{\beta}\rangle-\frac{2}{\pi}\ell_{\alpha}\delta_{\alpha\beta}=O(\ell_{\alpha}^{2}\ell_{\beta}^{2})$

where the constant of the $O$ notation depends only on $c_{0}>0$ with
$\ell_{\alpha},\ell_{\beta}\leq c_{0}$.

######  Corollary 3.5.

Given $c_{0}>0$, there is a two variable function d with the following
property. Given $l,a\in[0,c_{0}]$ such that $l>a\geq 0$. Let
$x,x^{\prime}\in\operatorname{Teich}(S)$ be such that $\ell_{\alpha}(x)\leq
l-a$ and $\ell_{\alpha}(x^{\prime})\geq l$. Then
$d_{\operatorname{WP}}(x^{\prime},x)\geq\textbf{d}(l,a)$.

###### Proof.

By Lemma 3.4 at $y\in\operatorname{Teich}(S)$ with $\ell_{\alpha}(y)\leq
c_{0}$,
$||\operatorname{grad}\ell_{\alpha}(y)||\leq(\frac{2}{\pi}\ell_{\alpha}(y)+O(\ell_{\alpha}(y))^{4})^{1/2}$
(*) where the $O$ notation constant depends only on $c_{0}$. Let $u$ be the WP
geodesic segment from $x$ to $x^{\prime}$, parametrized by arc-length. Let
$t^{*}$ be the first time that $\ell_{\alpha}(u(t))=l$. Then
$\ell_{\alpha}(u(t))\leq l$ for every $t\in[0,t^{*}]$. Using this bound and
integrating (*) we get

$a\leq|\ell_{\alpha}(u(t^{*}))-\ell_{\alpha}(u(0))|\leq\int_{0}^{t^{*}}||\operatorname{grad}\ell_{\alpha}(u(t))||dt\leq(\frac{2}{\pi}l+O(l)^{4})^{1/2}t^{*}.$

Hence
$t^{*}\geq\textbf{d}(l,a):=\frac{a}{\big{(}\frac{2}{\pi}l+O(l^{4})\big{)}^{1/2}}$.
∎

Using the estimates on pairings of the gradient of length-functions Wolpert
gives an asymptotic expansion for the WP metric near completion strata. These
expansion shows that there are asymptotic quasi-flats transverse to strata
corresponding to any pair of pinching curves. He also gives the following
estimate for the distance of a point in Teichmüller space and stratum.

######  Proposition 3.6.

(Corollary 4.10 of [Wol08]) Let $x\in\operatorname{Teich}(S)$ and $\sigma$ be
a multi-curve, then
$d_{\operatorname{WP}}(x,\mathcal{S}(\sigma))\leq\big{(}2\pi\sum_{\alpha\in\sigma}\ell_{\alpha}(x)\big{)}^{1/2}.$

Tangent cones of the Weil-Petersson completion of Teichmüller space: The
completion of Teichmüller space with the Weil-Petersson metric is a
$\operatorname{CAT}(0)$ space. Assigned to any point $p$ of a
$\operatorname{CAT}(0)$ space there is $AC_{p}$ the Alexandrov tangent cone
consisting of equivalence classes of geodesics $\zeta$ starting at the point
$p$. Two geodesics $\zeta$ and $\zeta^{\prime}$ starting at $p$ are equivalent
if their angle at $p$ in the sense of Alexandrov is equal to $0$. For more
detail about tangent cones see [BH99].

Given a multi-curve $\sigma$ on $S$ let $\chi$ be a full marking on
$S\backslash\sigma$ and consider the map

$\mathcal{L}(\zeta(t))=(\ell_{\alpha}^{1/2}(\zeta(t)),\ell_{\beta}^{1/2}(\zeta(t)))_{\alpha\in\sigma,\beta\in\chi}$

where $\zeta:[0,T]\to\overline{\operatorname{Teich}(S)}$ is a geodesic segment
with $\zeta(0)=p\in\mathcal{S}(\sigma)$. Then define
$\Lambda:AC_{p}\to\mathbb{R}^{+}$ by

$\Lambda(\zeta)=(2\pi)^{1/2}\frac{d}{dt}\Big{|}_{t=0}\mathcal{L}(\zeta(t)).$

Wolpert gives the following description of the WP Alexandrov tangent cone of
Teichmüller space at given point $p$, Theorem 4.18 of [Wol08].

######  Proposition 3.7.

(WP tangent cone) The map $\Lambda$ from the tangent cone of the WP metric at
$p$ to $\mathbb{R}_{\geq 0}^{|\sigma|}\times
T_{p}\operatorname{Teich}(S\backslash\sigma)$ is an isometry of tangent cones
with restriction of inner products. A WP geodesic $\zeta$ with $\zeta(0)=p$
and root length-function initial derivative
$\frac{d}{dt}\big{|}_{t=0}\ell_{\alpha}^{1/2}(\zeta(t))$ vanishing is
contained in the stratum $\\{\ell_{\alpha}=0\\}$,
$\mathcal{S}(\sigma)\subset\\{\ell_{\alpha}=0\\}$.

###  3.1. End invariant

In this subsection we recall the notion of end invariant for WP geodesics
introduced by Brock, Masur and Minsky in [BMM11].

######  Theorem 3.8.

(Convexity of length-functions)[Wol08] Given $\epsilon>0$ there is
$c=c(\epsilon)$ with the following property. Let
$g:(a,b)\to\operatorname{Teich}(S)$ be a WP geodesic parametrized by arc-
length and $\alpha\in\mathcal{C}_{0}(S)$. If for some $t\in(a,b)$
$\operatorname{inj}g(t)\geq\epsilon$ ($g(t)$ is a point in the
$\epsilon-$thick part of Teichmüller space), then we have

(3.1) $\ddot{\ell}_{\alpha}(g(t))\geq c\ell_{\alpha}(g(t)).$

Similar inequality holds for the length of any measured lamination
$\mathcal{L}$, which is

$\ddot{\ell}_{\mathcal{L}}(g(t))\geq c\ell_{\mathcal{L}}(g(t)).$

######  Remark 3.9.

The above estimates are local and only depend on the injectivity radius of the
surface $g(t)$.

######  Definition 3.10.

(Ending measured lamination) The weak∗ limit in $\mathcal{ML}(S)$ of any
weighted sequence of distinct Bers curves along a WP geodesic ray $r$ is an
ending measured lamination of $r$.

In [BMM11] the following notion of ending lamination for WP geodesic rays is
introduced, its existence relies on the convexity of length-functions along WP
geodesics and properties of $\operatorname{CAT}(0)$ spaces.

######  Definition 3.11.

(Ending Lamination) The union of pinching curves along a WP geodesic ray and
the geodesic laminations arising as supports of all ending measured
laminations of $r$ is the ending lamination of $r$.

######  Definition 3.12.

(End invariant of Weil-Petersson geodesics) To each open end of a geodesic
$g:(a,b)\to\overline{\operatorname{Teich}(S)}$ we associate an end invariant
which is a partial marking or a lamination. If the forward trajectory
$g|_{[0,b)}$ can be extended to $b$ such that
$g(b)\in\overline{\operatorname{Teich}(S)}$ then the forward end invariant
$\nu^{+}(g)$ is any Bers marking $\mu(g(b))$ ( there are finitely many of
them). Otherwise, $\nu^{+}(g)$ is the ending lamination of the forward
trajectory ray $g|_{[0,b)}$ which was defined above. We define the backward
end invariant $\nu^{-}(g)$ similarly by considering the backward trajectory
$g|_{(a,0]}$. We call the pair $(\nu^{-},\nu^{+})$, the end invariant of $g$.

Here we recall two properties of the ending measured laminations proved in
[BMM10]:

######  Lemma 3.13.

(Decreasing of length-functions along WP geodesic rays) Let $\mathcal{L}$ be
any ending measured lamination of a WP geodesic ray $r$, then
$\ell_{\mathcal{L}}(r(t))$ is a decreasing function.

######  Lemma 3.14.

Let $r_{n}\to r$ be a convergent sequence of rays in the WP visual sphere at
$x$. Then if $\mathcal{L}_{n}$ is any sequence of ending measured laminations
or weighted pinching curves for $r_{n}$, any representative
$\mathcal{L}\in\mathcal{ML}(S)$ of the limit of the projective classes
$[\mathcal{L}_{n}]$ in $\mathcal{PML}(S)$ has bounded length along the ray
$r$.

##  4\. Length-function control along uniformly bounded length Weil-Petersson
geodesics

In this section we study length-functions and twist parameters along sequences
of bounded length WP geodesic segments in the WP completion of Teichmüller
space.

In $\S$4.2 we will prove a modified version of Lemma 4.5 in [BMM11] about the
development of Dehn twists along sequences of uniformly bounded length WP
geodesic segments (Theorem 4.6). Corollaries 4.13 and 4.12 are somewhat
quantified versions of this theorem which provide us with a kind of twist
parameter versus length-function control along WP geodesic segments. This
control plays an important role in $\S$6 where we study the itinerary of WP
geodesics fellow traveling hierarchy paths.

The proof of Theorem 4.6 uses Wolpert’s characterization of limits of
sequences of uniformly bounded length WP geodesic segments in the Weil-
Petersson completion of Teichmüller space. In $\S$4.1 we state Wolpert’s
geodesic limit theorem and using suggestions of Jeff Brock will give an
improved version of it (Theorem 4.5). This improved version is crucial to
prove our results in $\S$4.2.

###  4.1. Limits of sequences of uniformly bounded length WP geodesic segments

In this subsection we provide a modified version of Wolpert’s geodesic limit
theorem. Given a multi-curve $\sigma$, denote by $\operatorname{tw}(\sigma)$
the subgroup of $\operatorname{Mod}(S)$ generated by the Dehn twists about the
curves in $\sigma$. Using the non-refraction property of the Weil-Petersson
completion strata (Theorem 3.2) and the fact that the quotient of any
$U_{\epsilon}(\sigma)$ by the action of $\operatorname{tw}(\sigma)$ is
compact, Wolpert provides the following characterization of the limits of
uniformly bounded length WP geodesic segments in the Teichmüller space. See
also [BMM11].

######  Theorem 4.1.

[Wol03] Let $\zeta_{n}:[0,T]\to\overline{\operatorname{Teich}(S)}$ be a
sequence of WP geodesic segments parametrized by arc-length of length $T$ in
the WP completion of the Teichmüller space. Then after possibly passing to a
subsequence there exist a partition of the interval $[0,T]$ by
$0=t_{0}<t_{1}<t_{2}<...<t_{k}<t_{k+1}=T$, simplices
$\sigma_{0},...,\sigma_{k+1}$, and simplices
$\tau_{i}=\sigma_{i-1}\cap\sigma_{i}$ ($i=1,...,k+1$) in
$\widehat{\mathcal{C}(S)}$ where $\sigma_{i}\subsetneq\tau_{i}$ for each
$1\leq i\leq k$, and a piecewise geodesic

$\hat{\zeta}:[0,T]\to\overline{\operatorname{Teich}(S)}$

with the following properties

  1. (1)

$\hat{\zeta}((t_{i-1},t_{i}))\subset\mathcal{S}(\tau_{i})$, for $i=1,...,k+1$,

  2. (2)

$\hat{\zeta}(t_{i})\in\mathcal{S}(\sigma_{i})$, for $i=0,...,k+1$,

  3. (3)

There are elements $\psi_{n}\in\operatorname{Mod}(S)$ and
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\tau_{i}\cup\tau_{i+1})$,
for $i=1,..,k$, such that after possibly passing to a subsequence
$\psi_{n}\circ\zeta_{n}(t)$ converges to $\hat{\zeta}(t)$ for every
$t\in[0,t_{1}]$ in $\overline{\operatorname{Teich}(S)}$ and for each
$i=1,..,k$, and $t\in[t_{i},t_{i+1}]$,

$\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}\circ\zeta_{n}(t)\to\hat{\zeta}(t)$

as $n\to\infty$. For convenience for each $i=0,1,...,k+1$ we define

(4.1)
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}$

  4. (4)

The elements $\psi_{n}$ are either trivial or unbounded and the elements
$\mathcal{T}_{i,n}$ are unbounded.

  5. (5)

The piecewise geodesic $\hat{\zeta}$ is the minimal length path in
$\overline{\operatorname{Teich}(S)}$ joining $\hat{\zeta}(0)$ to
$\hat{\zeta}(T)$ and intersecting the strata
$\mathcal{S}(\sigma_{1}),\mathcal{S}(\sigma_{2}),...,\mathcal{S}(\sigma_{k})$
in order.

The following two lemmas which were suggested to us by Jeff Brock help us to
considerably improve the above picture of limits of uniformly bounded length
WP geodesic segments (see Theorem 4.5).

######  Lemma 4.2.

Given a sequence of WP geodesic segments
$\zeta_{n}:[0,T]\to\operatorname{Teich}(S)$, let the simplices
$\tau_{i},\;i=1,...,k+1,$ be as in Theorem 4.1. Then
$\tau_{1}=...=\tau_{k+1}$. We denote

(4.2) $\hat{\tau}\equiv\tau_{i},\;\;i=1,...,k$

###### Proof.

Let the piecewise geodesic path $\hat{\zeta}:[0,T]\to\operatorname{Teich}(S)$,
the partition $0=t_{0}<t_{1}<...<t_{k+1}=T$ and simplices
$\sigma_{i},\;i=0,...,k+1$ be as in Theorem 4.1. Let
$\delta<\min_{i=1,...,k+1}\frac{t_{i}-t_{i-1}}{2}$. Let $0\leq i\leq k$. By
Theorem 4.1 (1),
$\hat{\zeta}|_{[t_{i}-\delta,t_{i})}\subset\mathcal{S}(\tau_{i})$ and
$\hat{\zeta}|_{(t_{i},t_{i}+\delta]}\subset\mathcal{S}(\tau_{i+1})$. Moreover
by Theorem 4.1 (5) the concatenation of $\hat{\zeta}|_{[t_{i}-\delta,t_{i})}$
and $\hat{\zeta}|_{[t_{i},t_{i}+\delta]}$ is the distance minimizing path in
$\overline{\operatorname{Teich}(S)}$ joining $\hat{\zeta}(t_{i}-\delta)$ to
$\hat{\zeta}(t_{i}+\delta)$ and intersecting $\mathcal{S}(\sigma_{i})$. Recall
that $\tau_{i+1}\subseteq\sigma_{i}$, so $\alpha\in\sigma_{i}$, then as
Wolpert shows on page 328 of [Wol08] the following equality of the one-sided
derivatives of the square root of the $\alpha-$length-function holds at
$t=t_{i}$,

(4.3)
$\frac{d}{dt}\Big{|}_{t=t_{i}^{+}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i},t_{i}+\delta]})=-\frac{d}{dt}\Big{|}_{t=t_{i}^{-}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i}-\delta,t_{i}]}).$

$\hat{\zeta}|_{[t_{i},t_{i}+\delta]}\subset\overline{\mathcal{S}(\tau_{i+1})}$,
so $\ell_{\alpha}^{1/2}(\hat{\zeta}(t))=0$ for all $t\in[t_{i},t_{i}+\delta]$
and thus
$\frac{d}{dt}|_{t=t_{i}^{+}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i},t_{i}+\delta]})=0$.
Then by (4.3)
$\frac{d}{dt}|_{t=t_{i}^{+}}\ell_{\alpha}^{1/2}(\hat{\zeta}|_{[t_{i},t_{i}+\delta]})=0$.
So by Proposition 3.7,
$\hat{\zeta}([t_{i}-\delta,t_{i}])\subset\overline{\mathcal{S}(\alpha)}$.
Moreover, again by Theorem 4.1 (1)
$\hat{\zeta}([t_{i}-\delta,t_{i}))\subset\mathcal{S}(\tau_{i})$. The last two
inclusions imply that $\alpha\in\tau_{i}$. This holds for every
$\alpha\in\tau_{i}$, so we conclude that $\tau_{i+1}\subseteq\tau_{i}$.
Exchanging the role of $\tau_{i}$ and $\tau_{i+1}$ a similar argument implies
that $\tau_{i}\subseteq\tau_{i+1}$. Thus $\tau_{i}=\tau_{i+1}$ for $i=1,...,k$
and we get $\tau_{1}=...=\tau_{k+1}$ as was desired. ∎

######  Lemma 4.3.

Let $\zeta_{n}:[0,T]\to\operatorname{Teich}(S)$ be a sequence of WP geodesic
segments parametrized by arc-length. Let $\mathcal{T}_{i,n},\;i=1,...,k,$ and
the simplices $\sigma_{i},\;i=0,...,k+1,$ be as in Theorem 4.1 and the simplex
$\hat{\tau}$ be as in (4.2). Let $1\leq i\leq k$. Suppose that
$\sigma_{i}-\hat{\tau}\neq\emptyset$ and $\gamma\in\sigma_{i}-\hat{\tau}$,
then the power of $\mathcal{D}_{\gamma}$ in $\mathcal{T}_{i,n}$ goes to
$\infty$ as $n\to\infty$.

###### Proof.

Let the elements of mapping class group $\varphi_{i,n},$ $i=1,...,k+1,$ be as
in (4.1). For $i=1,...,k$ define

$\sigma_{i,n}=\varphi_{i-1,n}^{-1}(\sigma_{i})$

also define the geodesic segments

$\zeta_{i,n}(t)=\varphi_{i-1,n}\circ\zeta_{n}(t)$ for $t\in[t_{i-1},t_{i+1}]$

where $0=t_{0}<t_{1}<...<t_{i}<t_{i+1}<...<t_{k+1}=T$ is the partition from
Theorem 4.1. We claim that

######  Claim 4.4.

There are $\epsilon_{1},\epsilon_{2}>0$ and $\delta>0$ depending only on the
sequence $\zeta_{n}$ such that for each $i=1,...,k$ and every $n$ sufficiently
large

  1. (i)

$\ell_{\gamma}(\zeta_{i,n}(t_{i}\pm\delta))\leq\epsilon_{2}$ for every
$\gamma\in\sigma_{i}$, and

  2. (ii)

The injectivity radius of the points $\zeta_{i,n}(t_{i}\pm\delta))$ is bounded
below by $\epsilon_{1}$ away from the collars of the curves in $\hat{\tau}$.

The claim follows from the limit picture of geodesics $\zeta_{n}$ and
continuity of length-functions. To see this, let
$\delta<\min_{i=1,...,k+1}\frac{t_{i}-t_{i-1}}{2}$. Fix $1\leq i\leq k$. By
Lemma 4.2 the two points $\hat{\zeta}(t_{i}\pm\delta)$ are in the stratum
$\mathcal{S}(\hat{\tau})$ and by Theorem 4.1 (2),
$\hat{\zeta}(t_{i})\in\mathcal{S}(\sigma_{i})$. Thus there are
$\epsilon^{\prime}_{1},\epsilon^{\prime}_{2}>0$ such that:

  1. (i’)

$\ell_{\gamma}(\hat{\zeta}(t_{i}\pm\delta))\leq\epsilon^{\prime}_{2}$ for
every $\gamma\in\sigma_{i}$, and

  2. (ii’)

The injectivity radius of the points $\zeta_{i,n}(t_{i}\pm\delta))$ is bounded
below by $\epsilon^{\prime}_{1}$ away from the collars of the curves in
$\hat{\tau}$.

Moreover, by Theorem 4.1 (3) for every $t\in[t_{i-1},t_{i}]$,
$\zeta_{i,n}(t)\to\hat{\zeta}(t)$ as $n\to\infty$. Thus by the continuity of
length-functions for $n$ sufficiently large the bounds (i) and (ii) at
$\zeta_{i,n}(t_{i}-\delta)$ for $\epsilon_{1}=\frac{\epsilon^{\prime}_{1}}{2}$
and $\epsilon_{2}=2\epsilon^{\prime}_{2}$ follow from the bounds in (i’) and
(ii’), respectively.

Since $\varphi_{i,n}=\mathcal{T}_{i,n}\circ\varphi_{i-1,n}$, by Theorem 4.1
(3) for every $t\in[t_{i},t_{i+1}]$,
$\mathcal{T}_{i,n}\circ\zeta_{i,n}(t)\to\hat{\zeta}(t)$ as $n\to\infty$. Then
by the continuity of length-functions from (i’) we obtain that
$\ell_{\gamma}(\mathcal{T}_{i,n}\circ\zeta_{i,n}(t_{i}+\delta))\leq\epsilon_{2}$
for every $\gamma\in\sigma_{i}$, and from (ii’) we obtain that the injectivity
radius of the point $\zeta_{i,n}(t_{i}-\delta))$ is bounded below by
$\epsilon^{\prime}_{1}$ away from the collars of the curves in $\hat{\tau}$.
It follows form Theorem 4.1 (3) and Lemma 4.2 that for each $n$,
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\hat{\tau})$. So applying
$\mathcal{T}_{i,n}$ does not change the length of every curve in $\sigma_{i}$
and injectvity radius. Thus the bounds (i) and (ii) at
$\zeta_{i,n}(t_{i}+\delta)$ follow from the ones we just established. This
finishes the proof of the claim.

We proceed to prove the lemma. Fix $\gamma\in\sigma_{i}-\hat{\tau}$. Let
$h_{1}:S\to\zeta_{i,n}(t_{i}-\delta)$ be the marking of the surface
$\zeta_{i,n}(t_{i}-\delta)$. Let
$\beta\in\mathcal{C}_{0}(S\backslash\hat{\tau})$ be a curve which has minimal
intersection number (1 or 2) with $\gamma$ and does not intersect any curve in
$\sigma_{i}-\gamma$ so that $h_{1}(\beta)$ has minimal length at
$\zeta_{i,n}(t_{i}-\delta)$, see Figure 1.

Figure 1.  $\beta\in\mathcal{C}_{0}(S\backslash\hat{\tau})$ is a curve which
has minimal intersection number one or two with $\gamma$ and does not
intersect any curve in $\sigma_{i}-\gamma$ so that has minimal length at
$\zeta_{i,n}(t_{i}-\delta)$.

Realize the curves $h_{1}(\beta)$ and $h_{1}(\gamma)$ as geodesics on
$\zeta_{i,n}(t_{i}-\delta)$. Denote the collar of $h_{1}(\gamma)$ by
$C(h_{1}(\gamma))$. Denote the length of each component of the boundary of the
collar by $C$. Further let $w$ be the width of the collar. Lifting the picture
to the universal cover the length of $h_{1}(\beta)\cap C(h_{1}(\gamma))$ is
bounded above by $w+2C$, see Figure 2. Further the length of $h_{1}(\beta)$
outside $C(h_{1}(\gamma))$ is bounded above by the diameter of
$\zeta_{i,n}(t_{i}-\delta)$ out side the collars of curves in $\hat{\tau}$
plus $2C$. By a compactness argument the diameter is bounded above by a
constant depending only on the injectivity radius of the surface out side the
collars of the curves in $\hat{\tau}$ and $C$. Then the length of
$h_{1}(\gamma)$ is bounded above by $w+2C+2C$ plus the diameter. Claim 4.4
(ii) provides the lower bound $\epsilon_{1}$ for the length of $\gamma$ and
consequently an upper bound for $C$ and $w$. Part (i) of the claim provides
the upper bound $\epsilon_{2}$ for the injectivity radius and consequently an
upper bound for the diameter. Thus there is $L$ depending only on
$\epsilon_{1},\epsilon_{2}$ so that

(4.4) $\ell_{\beta}(\zeta_{i,n}(t_{i}-\delta))\leq L.$

Let $h_{2}:S\to\zeta_{i,n}(t_{i}+\delta)$ be the marking of
$\zeta_{i,n}(t_{i}+\delta)$. $h_{2}(\beta)=h_{1}\circ\mathcal{T}_{i,n}(\beta)$
and $h_{2}(\gamma)=h_{1}(\gamma)$. Denote by $\mathcal{D}_{\gamma}$ the Dehn
twist about $\gamma$. Let $m_{i}$ be the power of $\mathcal{D}_{\gamma}$ in
$\mathcal{T}_{i,n}$. Realize $h_{2}(\gamma)$ and
$h_{2}(\mathcal{D}_{\gamma}^{m_{i}}(\beta))$ as geodesics. Lifting the picture
to the universal cover the length of $\mathcal{D}_{\gamma}^{m_{i}}(\beta)\cap
C(h_{2}(\gamma))$ is bounded above by
$|m_{i}|\ell_{\gamma}(\zeta_{i,n}(t_{i}+\delta))+w+2C$, see Figure 2. Further
its length outside the collar is bounded by the diameter of
$\zeta_{i,n}(t_{i}+\delta)$ out side the collars of the curves in $\hat{\tau}$
plus $2C$. Then the length of $\mathcal{D}_{\gamma}^{m_{i}}(\beta)$ is bounded
above by $w+|m_{i}|\ell_{\gamma}(\zeta_{i,n}(t_{i}+\delta))+2C+2C$ plus the
diameter. Suppose that $|m_{i}|$ is bounded by some $N>0$. Then as we saw
above Claim 4.4 gives us $L^{\prime}$ depending only on
$\epsilon_{1},\epsilon_{2}$ and $N$ so that

(4.5) $\ell_{\beta}(\zeta_{i,n}(t_{i}+\delta))\leq L^{\prime}.$

On the other hand, since $\ell_{\gamma}(\zeta_{i,n}(t_{i}))\to 0$ as
$n\to\infty$ and $\beta\pitchfork\gamma$, by the Collar lemma ($\S 4.1$ of
[Bus10]) we have that

(4.6) $\ell_{\beta}(\zeta_{i,n}(t_{i}))\to\infty$

as $n\to\infty$.

For $n$ sufficiently large, by (4.6),
$\ell_{\beta}(\zeta_{i,n}(t_{i}))>\max\\{L,L^{\prime}\\}$. But
$t_{i}-\delta<t_{i}<t_{i}+\delta$, then this bound, (4.4) and (4.5) violate
the convexity of the $\beta-$length-function along the WP geodesic segment
$\zeta_{i,n}$. This contradiction shows that the power of
$\mathcal{D}_{\gamma}$ in $\mathcal{T}_{i,n}$ is unbounded. ∎

Figure 2.  The lift of the collar of $h_{2}(\gamma)$ and the lift of
$h_{2}(\mathcal{D}_{\gamma}^{m_{i}}(\beta))$ to the universal cover. The
length of intersection of $h_{2}(\mathcal{D}_{\gamma}^{m_{i}}(\beta))$ and the
collar is bounded above by length of the red path. The segment $cd$ has length
$|m_{i}|\ell_{\gamma}(\zeta_{i,n}(t_{i}+\delta))$. The length of the segments
$ab$ and $ef$ are bounded by $C$ the length of each component of the boundary
of the collar. The length of the segments $bc$ and $de$ are $w$ the width of
the collar.

Here for the purpose of reference in this paper we state the following
strength version of the geodesic limit theorem which essentially contains the
properties listed in Theorem 4.1 modified to incorporate Lemmas 4.2 and 4.3.

######  Theorem 4.5.

(Geodesic Limit) Let $\zeta_{n}:[0,T]\to\overline{\operatorname{Teich}(S)}$ be
a sequence of WP geodesic segments of length $T$. Then after possibly passing
to a subsequence there exists a partition of the interval $[0,T]$ by
$0=t_{0}<t_{1}<...<t_{k}<t_{k+1}=T$, and simplices
$\sigma_{0},...,\sigma_{k+1}$ in $\widehat{\mathcal{C}(S)}$ such that
$\sigma_{i}\cap\sigma_{i+1}=\hat{\tau}$, for $i=0,...,k$, and a piecewise
geodesic

$\hat{\zeta}:[0,T]\to\overline{\operatorname{Teich}(S)}$

with the following properties

  1. (1)

$\hat{\zeta}((t_{i-1},t_{i}))\subset\mathcal{S}(\hat{\tau})$, for
$i=1,...,k+1$,

  2. (2)

$\hat{\zeta}(t_{i})\in\mathcal{S}(\sigma_{i})$, for $i=0,...,k+1$,

  3. (3)

There are elements $\psi_{n}\in\operatorname{Mod}(S)$ and
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\hat{\tau})$, for
$i=1,..,k$, such that for every $t\in[0,t_{1}]$, $\psi_{n}\circ\zeta_{n}(t)$
converges to $\hat{\zeta}(t)$, and for each $i=1,..,k$, and every
$t\in[t_{i},t_{i+1}]$,

$\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}\circ\zeta_{n}(t)\to\hat{\zeta}(t)$

as $n\to\infty$. For convenience for every $i=0,...,k+1$ we define

(4.7)
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{1,n}\circ\psi_{n}$

  4. (4)

The elements $\psi_{n}$ are either trivial or unbounded. Moreover, for any
$1\leq i\leq k$ and $\gamma\in\sigma_{i}$ the power of $\mathcal{D}_{\gamma}$
in the element $\mathcal{T}_{i,n}$ goes to $\infty$ as $n\to\infty$.

  5. (5)

The piecewise geodesic $\hat{\zeta}$ is the minimal length path in
$\overline{\operatorname{Teich}(S)}$ joining $\hat{\zeta}(0)$ to
$\hat{\zeta}(T)$ and intersecting the strata
$\mathcal{S}(\sigma_{1}),\mathcal{S}(\sigma_{2}),...,\mathcal{S}(\sigma_{k})$
in order.

By Lemma 4.2 $\hat{\tau}\equiv\sigma_{i}\cap\sigma_{i+1}$ for $i=0,...,k$.
Then part (1) follows from this and part (1) of Theorem 4.1. Part (4) follows
from part (4) in Theorem 4.1 and Lemma 4.3 .

###  4.2. Length-function versus twist parameter control

In this subsection we show that, roughly speaking, provided a lower bound for
the length of a curve $\gamma$ at the end points of a uniformly bounded length
WP geodesic segment $\zeta$, the higher Dehn twist about $\gamma$ forces
$\gamma$ to get shorter along $\zeta$ (Corollary 4.12). Moreover, in Corollary
4.13 we show that the shorter $\gamma$ gets along $\zeta$ the higher Dehn
twist develops about $\gamma$.

The main technical part of this subsection is the following modification of
Lemma 4.5 in [BMM11].

######  Theorem 4.6.

Given $\epsilon_{0},T$ and $s$ positive, let
$\zeta_{n}:[0,T_{n}]\to\operatorname{Teich}(S)$ be a sequence of Weil-
Petersson geodesic segments parametrized by arc-length of length $2s\leq
T_{n}\leq T$. Then given a sequence of curves $\alpha_{n}$ we have the
following:

  1. (1)

If there are subintervals $J_{n}\subseteq[s,T_{n}-s]$ such that

    1. (a)

$\sup_{t\in J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$, and

    2. (b)

$\inf_{t\in J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))\to 0$ as $n\to\infty$

then

$d_{\alpha_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\to\infty$

as $n\to\infty$.

  2. (2)

If

    1. (a)

$\sup_{t\in[0,T_{n}]}\ell_{\alpha_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$, and

    2. (b)

$d_{\alpha_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\to\infty$ as
$n\to\infty$

then

$\inf_{t\in[0,T_{n}]}\ell_{\alpha_{n}}(\zeta_{n}(t))\to 0$

as $n\to\infty$.

######  Remark 4.7.

Here we prove that the conditions in parts (1) and (2) hold for the sequence
of curves $\alpha_{n}$, then the conclusions hold for the sequence
$\alpha_{n}$ itself. In Lemma 4.5 of [BMM11] the conclusions are proved for
some sequence of curves $\beta_{n}$ such that each $\beta_{n}$ does not
overlap $\alpha_{n}$.

###### Proof.

Trimming the intervals slightly and changing the parameters $s$ and $T$ we may
assume that $T_{n}\equiv T$ for some $T\geq 2s$. After possibly passing to a
subsequence by Theorem 4.5, there exist a partition of $[0,T]$ with
$0=t_{0}<t_{1}<...<t_{k+1}=T$, simplices $\sigma_{i}$ for $i=0,...,k+1$ in
$\widehat{\mathcal{C}(S)}$ and a simplex $\hat{\tau}$, and a piecewise
geodesic path

$\hat{\zeta}:[0,T]\to\overline{\operatorname{Teich}(S)}$

so that $\hat{\zeta}([t_{i},t_{i+1}])$ is a geodesic segment in
$\mathcal{S}(\hat{\tau})$ joining the stratum $\mathcal{S}(\sigma_{i})$ to
$\mathcal{S}(\sigma_{i+1})$. Let
$\varphi_{i,n}=\mathcal{T}_{i,n}\circ...\circ\mathcal{T}_{i,1}\circ\psi_{n}$
be as in (4.7), where
$\mathcal{T}_{i,n}\in\operatorname{tw}(\sigma_{i}-\hat{\tau})$ and
$\psi_{n}\in\operatorname{Mod}(S)$ is either trivial or unbounded.

We start by setting up some notation. For each $i=0,...,k+1$ and $n\geq 1$ let

$\sigma_{i,n}=\varphi_{i,n}^{-1}(\sigma_{i})=\varphi_{i-1,n}^{-1}(\sigma_{i})$

be the pull backs of $\sigma_{i}$ to the $\zeta_{n}$ picture. For each
$i=1,...,k$ and $n\geq 1$ let

$\tau_{i,n}=\varphi_{i-1,n}^{-1}(\hat{\tau})$

For each $i=0,...,k+1$, choose a partial marking $\mu_{i}$ such that

  1. (1)

$\sigma_{i}\subset\operatorname{base}(\mu_{i})$, and

  2. (2)

$\mu_{i}$ restricts to a full marking of each connected component $Y\subseteq
S\backslash\hat{\tau}$ with complexity at least one.

For each $i=1,...,k+1$ and $n\geq 1$ define the pullback marking

$\mu_{i,n}^{-}=\varphi_{i-1,n}^{-1}(\mu_{i})$

and for each $i=0,...,k$ and $n\geq 1$ the pullback marking

$\mu_{i,n}^{+}=\varphi_{i,n}^{-1}(\mu_{i})$

Let $1\leq i\leq k$ and let $\gamma\in\sigma_{i}-\hat{\tau}$. In the following
three claims we will measure the twisting of these markings relative to
$\gamma_{n}=\varphi_{i,n}^{-1}(\gamma)$ and prove that

(4.8) $d_{\gamma_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T)))\to\infty$

as $n\to\infty$.

######  Claim 4.8.

$d_{\gamma_{n}}(\mu^{-}_{j,n},\mu^{+}_{j,n})$ is bounded for any $j=1,...,k$
with $j\neq i$.

First note that $\operatorname{base}(\mu^{+}_{j,n})$ and
$\operatorname{base}(\mu^{-}_{j,n})$ both contain

$\sigma_{j,n}=\varphi_{j,n}^{-1}(\sigma_{j})=\varphi_{j-1,n}^{-1}(\sigma_{j})$

Now we verify that

(4.9) $\gamma_{n}\notin\sigma_{j,n},\;\;\text{for any $j\neq i$}$

Otherwise, the length of $\gamma_{n}$ would converge to $0$ at
$\zeta_{n}(t_{i})$ and $\zeta_{n}(t_{j})$, and hence by the convexity of the
length-functions on all of $[t_{i-1},t_{i}]$ or $[t_{i},t_{i+1}]$ (the first
if $j<i$ and the second if $j>i$). So by Theorem 4.5 (1),
$\gamma\in\hat{\tau}$. But this contradicts the choice of
$\gamma\in\sigma_{i}-\hat{\tau}$. So (4.9) holds.

$\mu_{j,n}^{-}$ and $\mu_{j,n}^{+}$ restrict to a full marking on
$S\backslash\sigma_{j,n}$. So by (4.9) $\gamma_{n}$ intersects $\mu_{j,n}^{-}$
and $\mu_{j,n}^{+}$ nontrivially, and consequently
$\pi_{\gamma_{n}}(\mu_{j,n}^{\pm})$ is nonempty. Let
$\widetilde{\mathcal{T}}_{j,n}=\varphi_{j,n}^{-1}\circ\mathcal{T}_{j,n}\circ\varphi_{j,n}$,
then $\mu_{j,n}^{-}=\widetilde{\mathcal{T}}_{j,n}(\mu_{j,n}^{+})$.
Furthermore, since $\mathcal{T}_{j,n}\in\operatorname{tw}(\sigma_{j})$,
$\widetilde{\mathcal{T}}_{j,n}$ is an element of
$\operatorname{tw}(\sigma_{j,n})$. So $\mu_{j,n}^{+}$ differs from
$\mu_{j,n}^{-}$ by composition of Dehn twists about the curves in
$\sigma_{j,n}$. By the definition of annular subsurface coefficients the
annular subsurface coefficient of a sequence of curves intersecting both
$\mu^{-}_{j,n}$ and $\mu^{+}_{j,n}$ could go to $\infty$, only if they are in
$\sigma_{j,n}$. But by (4.9) it is not the case. The claimed bound follows
from this contradiction.

######  Claim 4.9.

$d_{\gamma_{n}}(\mu^{+}_{j,n},\mu^{-}_{j+1,n})$ is bounded for $j=1,...,k$.

$\mu_{j}$ and $\mu_{j+1}$ are full markings of $S\backslash\hat{\tau}$, where
their marking distance is some finite number. Hence we may connect them with a
finite sequence of full markings of $S\backslash\hat{\tau}$. Applying
$\varphi_{j,n}^{-1}$ to this sequence we obtain a sequence of the same length
connecting $\mu_{j,n}^{+}$ to $\mu_{j+1,n}^{-}$ through full markings of
$S\backslash\tau_{j+1,n}$. Moreover, $\gamma_{n}\notin\tau_{j+1,n}$ (because
$\gamma\notin\hat{\tau}$), so all of the markings in the connecting sequence
intersect $\gamma_{n}$ nontrivially. Any two consecutive markings in the
sequence differ by an elementary move and each elementary move increases the
$A(\gamma_{n})$ subsurface coefficient by at most one, then the claimed bound
follows.

######  Claim 4.10.

(4.10) $d_{\gamma_{n}}(\mu^{-}_{i,n},\mu^{+}_{i,n})\to\infty\;\;\text{as
$n\to\infty$}$

$\varphi_{i,n}(\mu^{-}_{i,n})=\mathcal{T}_{i,n}(\mu_{i})$, so after applying
$\varphi_{i,n}$ to all of the curves in the subsurface coefficient in (4.10)
we get

$d_{\gamma}(\mathcal{T}_{i,n}(\mu_{i}),\mu_{i})$

Now $\mu_{i}$ is a fixed marking which contains $\gamma$ as well as a
transversal curve for $\gamma$. By Theorem 4.5 (4) $\mathcal{T}_{i,n}$
contains an arbitrarily large power of $\mathcal{D}_{\gamma}$. So we obtain
the claimed bound.

Combining the bounds established in claims 4.8, 4.9 and 4.10 with the triangle
inequality the bound (4.8) follows. Having this bound we proceed by proving
our theorem.

Proof of part (1): We show that after possibly passing to a subsequence there
is an $1\leq i\leq k$ and a curve $\gamma\in\sigma_{i}-\hat{\tau}$, such that
$\alpha_{n}=\varphi_{i,n}^{-1}(\gamma)$. Part (1) then follows from (4.8).

Let $J_{n}\subset[s,T-s]$ be the subintervals in the statement of part (1).
Passing to a subsequence we may assume that $J_{n}$’s converge to a
subinterval $J$. Since each $J_{n}\subset[s,T-s]$, then $J\subset[s,T-s]$. For
each $i=0,...,k+1$,
$\varphi_{i,n}\circ\zeta_{n}|_{[t_{i},t_{i+1}]}\to\hat{\zeta}|_{[t_{i},t_{i+1}]}$
and $\varphi_{i,n}$ is isometry of the WP metric. So the length of
$\zeta_{n}(J_{n})$’s converge to the length of $\hat{\zeta}(J)$. Now since
$\zeta_{n}$’s and $\hat{\zeta}$ are (piece-wise) geodesics parametrized by
arc-length it follows that the length of the intervals $J_{n}$’s converge to
the length of the interval $J$. Now we show that the length of all $J_{n}$’s
are uniformly bounded below. For each $n\geq 1$, by 1(a),
$\ell_{\alpha_{n}}(\zeta_{n}(t))$ achieves the value $\epsilon_{0}$ in $J_{n}$
and by 1(b), for $n$ sufficiently large $\inf_{t\in
J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))\leq\frac{\epsilon_{0}}{2}$. Thus by
Corollary 3.5 the length of $J_{n}$ is at least
$\textbf{d}(\epsilon_{0},\frac{\epsilon_{0}}{2})>0$. This uniform lower bound
for the length of $J_{n}$ for all $n$ sufficiently large and the convergence
of the length of $J_{n}$’s to that of $J$ implies that $J$ has length at least
$\textbf{d}(\epsilon_{0},\frac{\epsilon_{0}}{2})$.

For each $n\geq 1$, let $t^{*}_{n}\in J_{n}$ be the time where $\inf_{t\in
J_{n}}\ell_{\alpha_{n}}(\zeta_{n}(t))$ is realized. There is an $0\leq i\leq
k+1$ such that for $n$ sufficiently large $t^{*}_{n}\in[t_{i},t_{i+1}]$ and
$t_{n}^{*}$ converge to some $t^{*}\in J\cap[t_{i},t_{i+1}]$.

First suppose that $t^{*}\neq t_{i},t_{i+1}$. By 1(b)
$\ell_{\alpha_{n}}(\zeta_{n}(t^{*}_{n}))\to 0$ as $n\to\infty$, so applying
$\varphi_{i,n}$ to $\ell_{\alpha_{n}}(\zeta_{n}(t_{n}^{*}))$ we have

(4.11)
$\ell_{\varphi_{i,n}(\alpha_{n})}(\varphi_{i,n}\circ\zeta_{n}(t_{n}^{*}))\to
0\;\text{as}\;n\to\infty.$

By Theorem 4.5 (3),
$\varphi_{i,n}\circ\zeta_{n}(t^{*}_{n})\to\hat{\zeta}(t^{*})$ as $n\to\infty$.
Moreover, the only curves with length $0$ at $\hat{\zeta}(t^{*})$ are the ones
in $\hat{\tau}$. Thus Theorem 3.1 (Continuity of length-functions) implies
that the curves in $\hat{\tau}$ are the only ones whose length at
$\varphi_{i,n}\circ\zeta_{n}(t^{*}_{n})$ converge to $0$. Then by (4.11) after
possibly passing to a subsequence $\varphi_{i,n}(\alpha_{n})=\beta$ for some
$\beta\in\hat{\tau}$. For each $l=0,...,k+1$, $\hat{\tau}\subseteq\sigma_{l}$,
so $\beta\in\sigma_{n}$. Now given $j=0,...,k+1$,
$\varphi_{j,n}\circ\varphi_{i,n}^{-1}$ is a composition of Dehn twists about
curves in $\sigma_{l}$, $l=j+1,...,i$ (see 4.7), so preserves $\beta$. Thus
$\varphi_{j,n}(\alpha_{n})=\beta$.

Given $t\in[0,T]$, $t\in[t_{j},t_{j+1}]$ for some $0\leq j\leq k$. By Theorem
4.5 (3) $\varphi_{j,n}(\zeta_{n}(t))\to\hat{\zeta}(t)$ as $n\to\infty$.
Moreover, $\ell_{\beta}(\hat{\zeta}(t))\equiv 0$ for all $t\in[0,T]$. Thus the
continuity of length-functions implies that
$\ell_{\beta}(\varphi_{j,n}(\zeta_{n}(t)))\to 0$ as $n\to\infty$. As we saw
above $\varphi_{j,n}(\alpha_{n})=\beta$, so
$\ell_{\varphi_{j,n}(\alpha_{n})}(\varphi_{j,n}(\zeta_{n}(t)))\to 0$ as
$n\to\infty$, then applying $\varphi_{j,n}^{-1}$ we have that
$\ell_{\alpha_{n}}(\zeta_{n}(t))\to 0$ as $n\to\infty$. But this contradicts
1(a).

So $t^{*}_{n}$ converges to either $t_{i}$ or $t_{i+1}$. Let $t^{*}_{n}\to
t_{i}(t_{i+1})$ as $n\to\infty$. Then
$\varphi_{i,n}(\zeta_{n}(t))\to\hat{\zeta}(t_{i})$
($\varphi_{i,n}(\zeta_{n}(t))\to\hat{\zeta}(t_{i+1})$) as $n\to\infty$. Note
that the only curves with length $0$ at
$\hat{\zeta}(t_{i})(\hat{\zeta}(t_{i+1}))$ are the ones in
$\sigma_{i}(\sigma_{i+1})$. Thus (4.11) and the convergence of length-
functions imply that
$\alpha_{n}=\varphi_{i,n}^{-1}(\gamma)(\varphi_{i+1,n}^{-1}(\gamma))$ for some
$\gamma\in\sigma_{i}-\hat{\tau}(\sigma_{i+1}-\hat{\tau})$, as was desired.

Proof of part (2): Suppose that for an $i$ with $\sigma_{i}\neq\emptyset$,
$\alpha_{n}\in\sigma_{i,n}$ for all $n\geq 1$. Applying $\varphi_{i,n}$ to
$\ell_{\alpha_{n}}(\zeta_{n}(t_{i}))$ we get
$\ell_{\varphi_{i,n}(\alpha_{n})}(\varphi_{i,n}(\zeta_{n}(t_{i})))$. By
Theorem 4.5 (3) we have that
$\varphi_{i,n}(\zeta_{n}(t_{i}))\to\hat{\zeta}(t_{i})$ as $n\to\infty$. Since
$\alpha_{n}\in\sigma_{i,n}:=\varphi_{i,n}^{-1}(\sigma_{i})$ we have that
$\varphi_{i,n}(\alpha_{n})\in\sigma_{i}$. Further the length of every curve in
$\sigma_{i}$ is $0$ at $\hat{\zeta}(t_{i})$. So the continuity of length-
functions implies that
$\ell_{\varphi_{i,n}(\alpha_{n})}(\varphi_{i,n}(\zeta_{n}(t_{i}))\to 0$ as
$n\to\infty$. Consequently, $\ell_{\alpha_{n}}(\zeta_{n}(t_{i}))\to 0$ as
$n\to\infty$. So the proof of part (2) would be complete if we show that for
some $i$ with $\sigma_{i}\neq\emptyset$, $\alpha_{n}\in\sigma_{i,n}$ for all
$n\geq 1$.

In contrary suppose that after possibly passing to a subsequence
$\alpha_{n}\notin\sigma_{i,n}$ for all $i=0,...,k+1$ and $n\geq 1$. Then
$\alpha_{n}$ intersects $\mu_{i,n}^{-}$ for $i=1,...,k+1$, and $\mu_{i,n}^{+}$
for $i=0,...,k$. For each $i$, let
$\widetilde{\mathcal{T}}_{i,n}=\varphi_{i,n}^{-1}\circ\mathcal{T}_{i,n}\circ\varphi_{i,n}$,
as before, as we saw earlier
$\widetilde{\mathcal{T}}_{i,n}\in\operatorname{tw}(\sigma_{i,n})$ and
$\mu_{i,n}^{-}=\widetilde{\mathcal{T}}_{i,n}(\mu_{i,n}^{+})$. So the only
annular subsurfaces of $\mu_{i,n}^{-}$ and $\mu_{i,n}^{+}$ which grow as
$n\to\infty$ are the ones with core curve in $\sigma_{i,n}$. Thus
$d_{\alpha_{n}}(\mu_{i,n}^{-},\mu_{i,n}^{+})$ is uniformly bounded for
$i=1,...,k$ and all $n\geq 1$.

Moreover, as we saw in the proof of Claim 4.9 the fact that for each
$i=0,...,k+1$, $\alpha_{n}$ intersects $\mu_{i,n}^{+}$ and $\mu_{i+1,n}^{-}$
implies that $d_{\alpha_{n}}(\mu_{i,n}^{+},\mu_{i+1,n}^{-})$ is uniformly
bounded for $i=1,...,k$ and all $n\geq 1$.

Combining the bounds form the above two paragraphs by the triangle inequality
we conclude that $d_{\alpha_{n}}(\mu_{0,n}^{-},\mu_{k,n}^{+})$ is uniformly
bounded above for all $n$. But $\mu(\zeta_{n}(0))=\mu_{0,n}^{-}$ and
$\mu(\zeta_{n}(T))=\mu_{k,n}^{+}$, so this bound contradicts assumption 2(b).

∎

######  Remark 4.11.

Fix a pants decomposition $P$ and let
$\big{(}\ell_{\gamma},\theta_{\gamma}\big{)}_{\gamma\in P}$ be a corresponding
Fenchel-Nielsen coordinates. Let $x\in\operatorname{Teich}(S)$ be a point with
$\theta_{\alpha}=\theta_{0}$ for some $\alpha\in P$. Let $\epsilon_{n}\to 0$.
Consider the following two sequences of WP geodesic segments:

  * •

Let $x_{n}$ ($n\geq 1$) be a point with all coordinates equal to that of $x$,
except for $\theta_{\alpha}(x_{n})=\theta_{0}+2\pi n$ and
$\ell_{\alpha}(x_{n})=\epsilon_{n}$. Then the length of the WP geodesic
segments $[x,x_{n}]$ is uniformly bounded and
$d_{\alpha}(\mu(x),\mu(x_{n}))\to\infty$ as $n\to\infty$.

  * •

Let $x_{n}$ ($n\geq 1$) be a point with all coordinates equal to that of $x$,
except for $\theta_{\alpha}=\theta_{0}$ and
$\ell_{\alpha}(x_{n})=\epsilon_{n}$. Then the length of WP geodesic segment
$[x,x_{n}]$ is uniformly bounded and $d_{\alpha}(\mu(x),\mu(x_{n}))=0$.

These two examples show that having a sequence of curves $\alpha_{n}$ whose
lengths converge to zero at the end points of a sequence of WP geodesic
segments, a-priori does not give any information about the growth of the
$A(\alpha_{n})$ subsurface coefficients of the Bers markings at the end points
of the geodesic segments. Thus the above theorem is the sharpest control of
length versus annular coefficient one could expect.

We are ready to prove the following two corollaries which are somewhat
quantified versions of Theorem 4.6. These corollaries provide us with a kind
of length-function versus twist parameter bounds over uniformly bounded length
WP geodesic segments which often will be used in $\S$6.

######  Corollary 4.12.

(large twist $\Longrightarrow$ short curve) Given $T,\epsilon_{0}$ and $N$
positive there is $\epsilon<\epsilon_{0}$ with the following property. Let
$\zeta:[0,T^{\prime}]\to\operatorname{Teich}(S)$ be a WP geodesic segment of
length $T^{\prime}\leq T$ such that

$\sup_{t\in[0,T^{\prime}]}\ell_{\gamma}(\zeta(t))\geq\epsilon_{0}$

If $d_{\gamma}(\mu(\zeta(0)),\mu(\zeta(T^{\prime})))>N$ then we have

$\inf_{t\in[0,T^{\prime}]}\ell_{\gamma}(\zeta(t))\leq\epsilon$

###### Proof.

The proof is by contradiction. Assume that the corollary does not hold. Then
there is a sequence of WP geodesic segments
$\zeta_{n}:[0,T_{n}]\to\operatorname{Teich}(S)$ parametrized by arc-length
with lengths $T_{n}\leq T$ and curves $\gamma_{n}\in\mathcal{C}_{0}(S)$, such
that

  1. (a)

$\sup_{t\in[0,T_{n}]}\ell_{\gamma_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$ for
every $n$,

  2. (b)

$d_{\gamma_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\to\infty$ as
$n\to\infty$,

and $\inf_{t\in[0,T_{n}]}\ell_{\gamma_{n}}(\zeta_{n}(t))>\epsilon$ for every
$n$. But this contradicts Theorem 4.6 (2).

∎

######  Corollary 4.13.

(short curve $\Longrightarrow$ large twist)

Given $\epsilon_{0},T,s$ and $\epsilon<\epsilon_{0}$ positive with $T>2s$,
there is an integer $N>0$ with the following property. Let
$\zeta:[0,T^{\prime}]\to\operatorname{Teich}(S)$ be a WP geodesic segment
parametrized by arc-length of length $T^{\prime}\in[2s,T]$,
$\gamma\in\mathcal{C}_{0}(S)$ and $J\subseteq[s,T^{\prime}-s]$ be a
subinterval such that:

$\sup_{t\in[0,T^{\prime}]}\ell_{\gamma}(\zeta(t))\geq\epsilon_{0}$

If $\inf_{t\in J}\ell_{\gamma}(\zeta(t))\leq\epsilon$, then

$d_{\gamma}(\mu(\zeta(0)),\mu(\zeta(T^{\prime})))>N$

###### Proof.

The proof is again by contradiction. Assume that the corollary does not hold.
Then there is a sequence of WP geodesic segments
$\zeta_{n}:[0,T_{n}]\to\operatorname{Teich}(S)$ parametrized by arc-length of
length $2s\leq T_{n}\leq T$ , $\gamma_{n}\in\mathcal{C}(S)$, subintervals
$J_{n}\subset[s,T_{n}-s]$ such that

  1. (a)

$\sup_{t\in J_{n}}\ell_{\gamma_{n}}(\zeta_{n}(t))\geq\epsilon_{0}$ for every
$n$,

  2. (b)

$\inf_{t\in J_{n}}\ell_{\gamma_{n}}(\zeta_{n}(t))\to 0$ as $n\to\infty$,

and $d_{\gamma_{n}}(\mu(\zeta_{n}(0)),\mu(\zeta_{n}(T_{n})))\leq N$ for every
$n$. But this contradicts Theorem 4.6 (1). ∎

######  Remark 4.14.

Note that these results have been proved using only compactness arguments in
the WP completion of Teichmüller space.

##  5\. Stable hierarchy paths

In this section we show that a certain class of hierarchy paths are stable in
the pants graph of the surface.

######  Definition 5.1.

($d-$stable subset) Given a function $d:\mathbb{R}^{\geq
1}\times\mathbb{R}^{\geq 0}\to\mathbb{R}^{\geq 0}$ a subset $\mathcal{Y}$ of a
metric space $\mathcal{X}$ is $d-$stable if for any $K\geq 1$ and $C\geq 0$
every $(K,C)-$quasi-geodesic $h$ with end points in $\mathcal{Y}$ is contained
in the $d(K,C)$ neighborhood of $\mathcal{Y}$. We call the function $d$ the
quantifier of the stability.

Here we summarize some of the results about stability of subsets of pants
graph of surfaces: Brock and Masur in [BM08] prove that when $\xi(S)=3$ the
pants graph of $S$ is strongly relatively hyperbolic with respect to the
quasi-flats corresponding to separating curves. The main ingredient of their
proof is that given a hierarchy path $\rho:[m,n]\to P(S)$ the subset of the
pants graph $X(\rho)=|\rho|\cup_{W}\\{P(W)\times P(W^{c})\\}$, where $W$ or
$W^{c}=S\backslash W$ is a component domain of $\rho$, is a stable subset of
the pants graph. Behrstock, Drutu and Mosher in [BDM09] study thick metric
spaces. These are metric spaces with rank at least $2$ where any two quasi-
flats are connected through a chain of quasi-flats with the property that any
two consecutive quasi-flats in the chain has coarse intersection of infinite
diameter. They show that thick metric spaces fail to be relatively hyperbolic
with respect to any collection of quasi-flats. Moreover, they observe that
$P(S)$ for $\xi(S)>3$ is a thick metric space and consequently is not
relatively hyperbolic with respect to any collection of quasi-flats. In
[BMM11] it is proved that hierarchy paths with bounded combinatorics end
points are stable.

Here we show that restricting the subsurfaces of a pair of partial markings or
laminations with subsurface coefficient bigger than a given $A>0$ to large
subsurfaces, implies stability of any hierarchy path $\rho$ between the pair
in the pants graph. We call such a pair $A-$narrow. Heuristically, these
hierarchy paths avoid quasi-flats in the pants graph corresponding to
separating multi-curves on the surface.

To be able to save considerable amount of work using results in the context of
$\Sigma-$hulls (see $\S$2.3) and present our results in a more general setting
we prove that $\Sigma-$hulls with the mentioned constraint on the subsurface
coefficients of their end points are stable. In $\S$5.2 we prove that for any
$\epsilon>F$ the $\Sigma_{\epsilon}-$hull of an $A-$narrow pair is
$d_{A}-$stable. Then the stability of hierarchy paths between the $A-$narrow
pair follows from the fact that the Hausdorff distance of a hierarchy path
between an $A-$narrow pair and the $\Sigma_{\epsilon}-$hull ($\epsilon$
sufficiently large) of the pair is bounded depending only on $A$ and
$\epsilon$. This is proved in Theorem 5.5.

###  5.1. Narrow pairs

In this subsection first we introduce the notion of an A-narrow pair of
makings or laminations. Then we will show that any hierarchy path between a
narrow pair and the $\Sigma_{\epsilon}-$hull ($\epsilon>0$ is sufficiently
large) of the pair have finite Hausdorff distance depending only on $A$ and
$\epsilon$.

######  Definition 5.2.

(Large subsurface) A connected essential subsurface $Z\subseteq S$ is called
large if any connected component of $S\backslash Z$ is either an annulus or a
three holed sphere.

######  Remark 5.3.

If $Z$ is not connected or $S\backslash Z$ has connected components other than
annuli and three holed spheres then $Z$ is not a large subsurface.

######  Definition 5.4.

($A-$narrow) A pair of partial markings or laminations $(\mu^{-},\mu^{+})$ is
called $A-$narrow if every non-annular subsurface $Z\subseteq S$ with the
property that

$d_{Z}(\mu^{-},\mu^{+})>A$

is a large subsurface of $S$.

Recall the constants $M_{1},M_{2}$ from Theorem 2.13 and $B_{0}$ from Theorem
2.8. We fix the constant $M=M_{1}+B_{0}+4$ in this subsection. Note that since
$M_{1}\geq 2M_{2}$, $M\geq M_{2}$.

######  Theorem 5.5.

($\Sigma-$hull of narrow pair)

Given $\epsilon>M$ and $A>4M+2\epsilon+12$, there is a constant
$\Delta=\Delta(A,\epsilon)$ with the following property. Given an $A-$narrow
pair $(\mu^{-},\mu^{+})$, the Hasudorff distance of the
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ and any hierarchy path between
$(\mu^{-},\mu^{+})$ is less than $\Delta$.

###### Proof.

Let $\rho:[m,n]\to P(S)$ be a hierarchy path with $\rho(m)=\mu^{-}$ and
$\rho(n)=\mu^{+}$. Let $X\subseteq S$ be a non-annular subsurface. Let
$i\in[m,n]$. By Theorem 2.13 (5), there is
$y\in\operatorname{hull}_{X}(\mu^{-},\mu^{+})$ such that $d_{X}(\rho(i),y)\leq
M_{2}<\epsilon$ (by the assumption of the theorem on the value of $\epsilon$,
$\epsilon>M_{2}$). Thus by the definition of $\Sigma_{\epsilon}-$hull we have
that $\rho(i)\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$. Consequently

$|\rho|\subset\Sigma_{\epsilon}(\mu^{-},\mu^{+})$

We proceed to prove that $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is contained in
a neighborhood of $|\rho|$.

######  Lemma 5.6.

Given $\epsilon>M$ and $A>4M+2\epsilon+12$, there is $d=d(A,\epsilon)$ with
the property that given $P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ there is
$\hat{j}\in[m,n]$ such that for every non-annular subsurface $X\subseteq S$ we
have

(5.1) $d_{X}(\rho(\hat{j}),P)\leq d$

###### Proof.

Let $X\subseteq S$ be a non-annular subsurface.
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, so there is
$x_{X}\in\operatorname{hull}_{X}(\mu^{-},\mu^{+})$ such that
$d_{X}(P,x_{X})\leq\epsilon$. By Theorem 2.13 (5) there is a vertex
$x^{\prime}_{X}\in\pi_{X}(|\rho|)$ ($|\rho|=\bigcup_{i\in[m,n]}\rho(i)$) with
$d_{X}(x_{X},x^{\prime}_{X})\leq M$. So by the triangle inequality,
$d_{X}(x^{\prime}_{X},P)\leq M+\epsilon$. Let $j\in[m,n]$ be such that
$\pi_{X}(\rho(j))\supset x^{\prime}_{Y}$, then

(5.2) $d_{X}(\rho(j),P)\leq M+\epsilon.$

Let $e=2A$. Then the subset of parameters
$\mathcal{E}_{X}(P)=\\{i\in[m,n]:d_{X}(P,\rho(i))\leq e\\}$ is non-empty.
Because any $j$ as above is in $\mathcal{E}_{X}$. Denote the minimum and
maximum of the set $\mathcal{E}_{X}(P)$ by $e_{X}^{-}$ and $e_{X}^{+}$
respectively. Let $E_{X}:=E_{X}(P)=[e_{X}^{-},e_{X}^{+}]$.

Recall the interval $J_{X}$ from Theorem 2.13 (1).

######  Claim 5.7.

Let $X$ be a component domain of $\rho$. Then

(5.3) $E_{X}\cap J_{X}\neq\emptyset$

$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, so there is
$x_{X}\in\operatorname{hull}_{X}(\mu^{-},\mu^{+})$ such that
$d_{X}(P,x_{X})\leq\epsilon$. $X$ is a component domain of $\rho$, so by
Theorem 2.13 (4) there is $x^{\prime}_{X}\in g_{X}$ such that
$d_{X}(x_{X},x^{\prime}_{X})\leq M_{2}$. Then by the triangle inequality we
have $d_{X}(P,x^{\prime}_{X})\leq M+\epsilon$. Let $j\in J_{X}$ be such that
$\rho(j)\supset x^{\prime}_{X}$. Then

(5.4) $d_{X}(\rho(j),P)\leq M+\epsilon.$

Now since $M+\epsilon>e$ we have that $j\in E_{X}$. The claim is proved.

Let $j\in E_{X}$. Suppose that $X$ is a component domain of $\rho$. Let
$v=\rho(j)\cap g_{X}$, $u=\rho(e_{X}^{-})\cap g_{X}$ and
$w=\rho(e_{X}^{+})\cap g_{X}$. Then by Theorem 2.13 (3) (Monotonicity), $u\leq
v\leq w$ as vertices along $g_{X}$. Furthermore, since
$d_{X}(P,\rho(e_{X}^{-}))\leq e$ we have $d_{X}(P,u)\leq e+1$ and since
$d_{X}(P,\rho(e_{X}^{+}))\leq e$ we have $d_{X}(P,w)\leq e+1$. Then by the
triangle inequality

$d_{X}(u,w)\leq 2e+2+\operatorname{diam}_{X}(P)\leq 2e+2+1.$

So either $d_{X}(u,v)\leq e+2$ or $d_{X}(v,w)\leq e+2$. Suppose that
$d_{X}(u,v)\leq e+2$. Then since $d_{X}(P,u)\leq e$ and $\rho(j)\supset v$, by
the triangle inequality we get

(5.5) $d_{X}(\rho(j),P)\leq 2e+2$

Similarly, we get the above bound assuming that $d_{W}(v,w)\leq e+2$.

Now suppose that $X$ is not a component domain of $\rho$. Then
$d_{X}(\mu^{-},\mu^{+})\leq M$. Furthermore, by Theorem 2.13 (5) the Hausdorff
distance of $\pi_{X}(|\rho|)$ and $\operatorname{hull}_{X}(\mu^{-},\mu^{+})$
is bounded above by $M$. Therefore, $\operatorname{diam}_{X}(|\rho|)\leq
3M+2$. Thus

(5.6) $d_{X}(\rho(j),P)\leq e+3M+2$

To prove the lemma it suffices to show that

(5.7) $\bigcap_{\begin{subarray}{c}X\subseteq S\\\ \text{non-
annular}\end{subarray}}E_{X}\neq\emptyset.$

To see this let $\hat{j}\in\bigcap_{\begin{subarray}{c}X\subseteq S\\\
\text{non-annular}\end{subarray}}E_{X}$. Then by the bounds (5.5) and (5.6)
for every non-annular subsurface $X\subseteq S$,

$d_{X}(\rho(\hat{j}),P)\leq\max\\{2e+2,e+3M+2\\},$

which is the desired bound in (5.1).

Our strategy to prove that (5.7) holds is to verify that for any two non-
annular subsurfaces $Y$ and $W$ we have that

(5.8) $E_{Y}\cap E_{W}\neq\emptyset.$

Then Helly’s Theorem in one dimension (see [Eck93]) implies that the
intersection of all of the intervals
$\\{E_{X}\\}_{\begin{subarray}{c}X\subseteq S\\\ \text{non-
annular}\end{subarray}}$ is nonempty.

If $d_{Y}(\mu^{-},\mu^{+})\leq A$, then
$\operatorname{diam}_{Y}(\operatorname{hull}_{Y}(\mu^{-},\mu^{+}))\leq A+2$.
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ so there is
$x_{Y}\in\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ with
$d_{Y}(P,x_{Y})\leq\epsilon$. By Theorem 2.13 (5) for every $i\in[m,n]$ there
is $y\in\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ such that
$d_{Y}(\rho(i),y)\leq M_{2}$. By the bound on the diameter of the hull
$d_{Y}(x_{Y},y)\leq A+2$. The last three bounds combined by the triangle
inequality give us

$d_{Y}(\rho(i),P)\leq\epsilon+M_{2}+A+2\leq e.$

Thus $E_{Y}=[m,n]$, which obviously intersects $E_{W}$. If
$d_{W}(\mu^{-},\mu^{+})\leq A$, similarly we can conclude that $E_{W}=[m,n]$,
which implies that $E_{W}\cap E_{Y}\neq\emptyset$. Therefore, in the rest of
the proof we may assume that

  * •

$d_{Y}(\mu^{-},\mu^{+})>A$, and

  * •

$d_{W}(\mu^{-},\mu^{+})>A$.

In particular, since $A>M$ both $Y$ and $W$ are component domains of $\rho$.

Now consider the following collection of subsurfaces:

  * •

$\mathfrak{L}:=\mathfrak{L}_{A}(\mu^{-},\mu^{+})=\\{X\subseteq S\;\;\text{non-
annular subsurface}:d_{X}(\mu^{-},\mu^{+})>A\\}$

For each $X\in\mathfrak{L}$ define

  * •

$i_{X}^{-}=\max\\{i\in[m,n]:d_{X}(\rho(i),\mu^{-})\leq M\\}$ and

  * •

$i_{X}^{+}=\min\\{i\in[m,n]:d_{X}(\rho(i),\mu^{+})\leq M\\}$.

######  Claim 5.8.

Let $J_{X}=[j_{X}^{-},j_{X}^{+}]$. Then $j_{X}^{-}\leq i_{X}^{-}\leq
i_{X}^{+}\leq j_{X}^{+}$ and we may write

(5.9) $[i_{X}^{-},i_{X}^{+}]\subseteq J_{X}$

By Theorem 2.13 (4) $d_{X}(\mu^{-},\rho(j_{X}^{-}))\leq M$ and
$d_{X}(\mu^{+},\rho(j_{X}^{+}))\leq M$. So we have that $i_{X}^{-}\geq
j_{X}^{-}$ and $i_{X}^{+}\leq j_{X}^{+}$. Then since $J_{X}$ is an interval,
$i_{X}^{-},i_{X}^{+}\in J_{X}$. Now we show that $i_{X}^{-}\leq i_{X}^{+}$. As
we just said, $d_{X}(\rho(j_{X}^{-}),\mu^{-})\leq M$. By the definition of
$i_{X}^{-}$, $d_{X}(\rho(i_{X}^{-}),\mu^{-})\leq M$. The last two inequalities
combined by the triangle inequality imply that

(5.10) $d_{X}(\rho(i^{-}_{X}),\rho(j_{X}^{-}))\leq 2M+2.$

Similarly we have that

(5.11) $d_{X}(\rho(i^{+}_{X}),\rho(j_{X}^{+}))\leq 2M+2.$

Furthermore, $d_{X}(\mu^{-},\mu^{+})>A>6M+12$, so by the bounds
$d_{X}(\mu^{-},\rho(j^{-}))\leq M$ and $d_{X}(\mu^{+},\rho(j^{+}))\leq M$ we
have that

(5.12) $d_{X}(\rho(j_{X}^{-}),\rho(j_{X}^{+}))>4M+12.$

Let $u=\rho(j_{X}^{-})\cap g_{X}$, $w=\rho(j_{X}^{+})\cap g_{X}$,
$v=\rho(i_{X}^{-})\cap g_{X}$ and $v^{\prime}=\rho(i_{X}^{+})\cap g_{X}$. By
(5.10),

$d_{X}(u,v)\leq
d_{X}(\rho(i_{X}^{-}),\rho(j_{X}^{-}))+\operatorname{diam}_{X}(\rho(i_{X}^{-}))+\operatorname{diam}_{X}(\rho(j_{X}^{-}))\leq
2M+4,$

by (5.11) $d_{X}(w,v^{\prime})\leq 2M+4$ and by (5.12) $d_{X}(u,w)\geq 4M+10$.
Then since $u<w$ as vertices along $g_{X}$ the last there inequalities imply
that $v<v^{\prime}$. Then Theorem 2.13 (3) (Monotonicity) implies that
$i_{X}^{-}\leq i_{X}^{+}$.

######  Claim 5.9.

Let $X\in\mathfrak{L}$. If $i\in[m,i_{X}^{-}]$ then
$d_{X}(\rho(i),\mu^{-})\leq 2M+3$. If $i\in[i_{X}^{+},n]$ then
$d_{X}(\rho(i),\nu^{+})\leq 2M+3$.

We prove the first part of the claim. The proof of the second part is similar.
Let $J_{X}=[j_{X}^{-},j_{X}^{+}]$. By Claim 5.8, $j_{X}^{-}\leq i_{X}^{-}$. If
$i\leq j_{X}^{-}$ then by Theorem 2.13 (4) we have $d_{X}(\mu^{-},\rho(i))\leq
M<2M+3$, which is the desired bound. Otherwise, $j_{X}^{-}\leq i\leq
i_{X}^{-}$. Note that $j_{X}^{-},i,i_{X}^{-}\in J_{X}$. Then let
$u=\rho(j_{X}^{-})\cap g_{X},v=\rho(i)\cap g_{X}$ and $w=\rho(i_{X}^{-})\cap
g_{X}$. Then by (5.10) we have

$d_{X}(u,w)\leq
2M+2+\operatorname{diam}_{X}(\rho(j_{X}^{-}))+\operatorname{diam}_{X}(\rho(i_{X}^{-}))=2M+4.$

Moreover, since $j_{X}^{-}\leq i\leq i_{X}^{-}$, by Theorem 2.13 (3)
(Monotonicity) $u\leq v\leq w$ as vertices along the geodesic
$g_{X}\subset\mathcal{C}(X)$. So by the above inequality either
$d_{X}(u,v)\leq M+2$ or $d_{X}(v,w)\leq M+2$. The former inequality implies
that $d_{X}(\rho(j_{X}^{-}),\rho(i))\leq M+2$. This inequality and
$d_{X}(\rho(j_{X}^{-}),\mu^{-})\leq M$ (Theorem 2.13 (4)) combined by the
triangle inequality imply that $d_{X}(\rho(i),\mu^{-})\leq
2M+2+\operatorname{diam}_{X}(\rho(j_{X}^{-}))\leq 2M+3$. The later inequality
implies that $d_{X}(\rho(i),\rho(j_{X}^{-}))\leq M+2$. This inequality and
$d_{X}(\mu^{-},\rho(j_{X}^{-}))\leq M$ (Theorem 2.13 (4)) combined by the
triangle inequality imply that $d_{Y}(\rho(i),\mu^{-})\leq 2M+3$. The first
part of the claim is proved.

Given $Y,W\in\mathfrak{L}$ the $A-$narrow condition implies that either

  1. (1)

$Y\pitchfork W$,

  2. (2)

$Y\supsetneq W$, or

  3. (3)

$Y\subsetneq W$.

In what follows we discuss these three cases and in each case verify that
(5.8) holds.

Case 1: $Y\pitchfork W$.

$Y,W\in\mathfrak{L}$ and $A>4M$. So $d_{Y}(\mu^{-},\mu^{+})>4M$ and
$d_{W}(\mu^{-},\mu^{+})>4M$. Then by Proposition 2.15 either $Y<W$ or $W<Y$
(not both). Assume that $Y<W$. $W<Y$ can be treated similarly. Then by the
proposition we have the following two inequalities

(5.13) $\displaystyle d_{Y}(\mu^{-},\partial{W})$ $\displaystyle>$
$\displaystyle M,\;\text{and}$ (5.14) $\displaystyle
d_{W}(\mu^{+},\partial{Y})$ $\displaystyle>$ $\displaystyle M.$

We proceed to discuss the following three subcases depending on the values of
$d_{Y}(P,\mu^{-})$ and $d_{Y}(P,\mu^{+})$. In each case we verify that (5.8)
holds.

Case 1.1:

(5.15) $d_{Y}(P,\mu^{-})\leq 3M+\epsilon+4.$

We show that the following inclusion of intervals holds

(5.16) $E_{Y}\supseteq[m,i_{Y}^{-}]$

Let $i\in[m,i^{-}_{Y}]$, then by the first part of Claim 5.9 we have
$d_{Y}(\rho(i),\mu^{-})\leq 2M+3$. This inequality and (5.15) combined by the
triangle inequality give us

$d_{Y}(P,\rho(i))\leq 5M+\epsilon+7+\operatorname{diam}_{Y}(\mu^{-})\leq
5M+\epsilon+9$

Furthermore, $e=2A>5M+\epsilon+9$, so by the definition of $E_{Y}$ we have
$i\in E_{Y}$.

Now we show that the inclusion of intervals

(5.17) $E_{W}\supseteq[m,i_{W}^{-}]$

holds. To see this, note that since $Y\in\mathfrak{L}$ we have
$d_{Y}(\mu^{-},\mu^{+})>A>4M+2\epsilon+12$. By (5.14) and Theorem 2.8
(Behrstock Inequality) we have, $d_{Y}(\partial{W},\mu^{+})\leq M$. The last
two bounds and (5.15) combined by the triangle inequality imply that
$d_{Y}(P,\partial{W})>\epsilon>M$. Thus by the Behrstock inequality we have

(5.18) $d_{W}(\partial{Y},P)\leq M.$

By (5.13) and the Behrstock inequality we have that
$d_{W}(\partial{Y},\mu^{-})\leq M$. This inequality and (5.18) combined by the
triangle inequality give us

(5.19) $d_{W}(P,\mu^{-})\leq 2M+1$

Let $i\in[m,i_{W}^{-}]$, by the first part of Claim 5.9,
$d_{W}(\mu^{-},\rho(i))\leq 2M+3$. Combining this inequality and (5.19) by the
triangle inequality we get

$d_{W}(P,\rho(i))\leq 4M+4.$

Now since $e>4M+4$, by the definition of $E_{W}$, $i\in E_{W}$. So we conclude
that $E_{W}\supseteq[m,i_{W}^{-}]$.

The inclusion of intervals (5.16) and (5.17) together imply that $E_{W}\cap
E_{Y}\neq\emptyset$.

Case 1.2:

(5.20) $\min\\{d_{Y}(P,\mu^{-}),d_{Y}(P,\mu^{+})\\}>3M+\epsilon+4$

We show that

(5.21) $E_{Y}\cap[i_{Y}^{-},i_{Y}^{+}]\neq\emptyset$

To see this, note that $P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, then by
(5.4), there is $i\in[m,n]$ such that

(5.22) $d_{Y}(P,\rho(i))\leq M+\epsilon$

Now since $e>M+\epsilon$, by the definition of $E_{Y}$, $i\in E_{Y}$.

By (5.20), $d_{Y}(P,\mu^{-})>3M+\epsilon+4$. Combining this inequality and
(5.22) by the triangle inequality we have that
$d_{Y}(\rho(i),\mu^{-})>2M+4-\operatorname{diam}_{Y}(P)=2M+3$. Then by the
contrapositive of the first part of Claim 5.9 we conclude that $i>i_{Y}^{-}$.
By (5.20), $d_{Y}(P,\mu^{+})>3M+\epsilon+4$. Combining this inequality and
(5.22) by the triangle inequality we have that $d_{Y}(\rho(i),\mu^{+})>2M+3$.
Then by the contrapositive of the second part of Claim 5.9 we conclude that
$i<i_{Y}^{+}$. Therefore $i\in[i_{Y}^{-},i_{Y}^{+}]$.

Now we show that the inclusion of intervals

(5.23) $E_{W}\supseteq[m,i_{W}^{-}]$

holds. By (5.14) and Theorem 2.8 (Behrstock Inequality),
$d_{Y}(\partial{W},\mu^{+})\leq M$. Moreover, by (5.20),
$d_{Y}(P,\mu^{+})>3M+\epsilon+4$. These two bounds combined by the triangle
inequality give us
$d_{Y}(P,\partial{W})>2M+\epsilon+4-\operatorname{diam}_{Y}(\mu^{+})>M$.
Therefore, by the Behrstock inequality

(5.24) $d_{W}(P,\partial{Y})\leq M$

Having (5.24), the rest of the proof of the inclusion of intervals (5.23)
follows from exact the same lines given after (5.18) to prove the inclusion of
intervals (5.17).

######  Claim 5.10.

$i_{Y}^{+}\leq i_{W}^{-}$.

By (5.14) and the Behrstock inequality, $d_{Y}(\partial{W},\mu^{+})\leq M$. By
(5.9) $i_{W}^{-}\in J_{W}$, so $\rho(i_{W}^{-})\supset\partial{W}$. Thus
$d_{Y}(\rho(i_{W}^{-}),\mu^{+})\leq M$. Then since $i_{Y}^{+}$ is the maximal
time such that $d_{Y}(\rho(i),\mu^{+})\leq M$ we have $i_{Y}^{+}\leq
i_{W}^{-}$.

By Claim 5.10, $[i_{Y}^{-},i_{Y}^{+}]\subseteq[m,i_{W}^{-}]$. Then by (5.23),
$[i_{Y}^{-},i_{Y}^{+}]\subseteq E_{W}$. So (5.21) implies that $E_{Y}\cap
E_{W}\neq\emptyset$.

Case 1.3:

(5.25) $d_{Y}(P,\mu^{+})\leq 3M+\epsilon+4.$

(5.25) and the second part of Claim 5.9 using an argument similar to the one
for the proof of (5.16) in Case 1.1 imply that

(5.26) $E_{Y}\supseteq[i_{Y}^{+},n]$

For every $j\in J_{W}$, $\rho(j)\supset\partial{W}$. By (5.14) and the
Behrstock inequality, $d_{Y}(\partial{W},\mu^{+})\leq M$. So
$d_{Y}(\rho(j),\mu^{+})\leq M$. Thus by the definition of $i_{Y}^{+}$, $j\geq
i_{Y}^{+}$. Therefore $J_{W}\subseteq[i_{Y}^{+},n]$. Then by (5.26)
$J_{W}\subseteq E_{Y}$. Moreover, by (5.3) $J_{W}\cap E_{W}\neq\emptyset$. So
we conclude that $E_{W}\cap E_{Y}\neq\emptyset$.

Case 2: $W\subsetneq Y$.

Recall the constant $e=2A$. We consider the following two subcases depending
on the value of $d_{Y}(\partial{W},P)$.

Case 2.1: $d_{Y}(\partial{W},P)\leq e$.

Let $i\in J_{W}$, by Theorem 2.13 (1) $\partial{W}\subset\rho(i)$, so
$d_{Y}(\rho(i),P)\leq d_{Y}(\partial{W},P)\leq e$. Recall that
$E_{Y}=\\{i:d_{Y}(\rho(i),P)\leq e\\}$, thus $J_{W}\subseteq E_{Y}$. Moreover,
by (5.3) $J_{W}\cap E_{W}\neq\emptyset$. Thus $E_{W}\cap E_{Y}\neq\emptyset$.

Case 2.2:

(5.27) $d_{Y}(\partial{W},P)>e.$

$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$, then as we saw in the paragraph
before (5.4), there is $x_{Y}\in\operatorname{hull}_{Y}(\mu^{-},\mu^{+})$ such
that $d_{Y}(x_{Y},P)\leq\epsilon$ and by Theorem 2.13 (5), there is
$x^{\prime}_{Y}$ on $g_{Y}$ such that $d_{Y}(x^{\prime}_{Y},x_{Y})\leq M$.
Then by the triangle inequality,

(5.28) $d_{Y}(x^{\prime}_{Y},P)\leq M+\epsilon.$

Let $h$ be a geodesic in $\mathcal{C}(Y)$ connecting $\pi_{Y}(P)$ to
$x^{\prime}_{Y}$. We claim that $h$ does not intersect the $1-$neighborhood of
$\partial{W}$. Otherwise, there is a vertex $z\in h$ with
$d_{Y}(z,\partial{W})\leq 1$, see Figure 3. Then we have

$\displaystyle d_{Y}(P,\partial{W})$ $\displaystyle\leq$ $\displaystyle
d_{Y}(P,z)+d_{Y}(z,\partial{W})$ $\displaystyle\leq$ $\displaystyle
d_{Y}(P,x^{\prime}_{Y})+1\leq(M+\epsilon)+1$

The first inequality is the triangle inequality and the second inequality
follows since $d_{Y}(P,z)\leq d_{Y}(P,x^{\prime}_{Y})$ (see Figure 3) and
$d_{Y}(z,\partial{W})\leq 1$. The third inequality holds by (5.28). But this
upper bound contradicts the lower bound (5.27) given as the assumption of Case
2.2.

Figure 3.  Case 2.2: If $h$ intersects the $1-$neighborhood of $\partial{W}$
then the distance between $\pi_{Y}(\partial{W})$ and $\pi_{Y}(P)$ would be
less than the lower bound in the assumption of Case 2.2.

By the above claim $\partial{W}$ intersects every vertex of $h$, so Theorem
2.6 (Bounded Geodesic Image Theorem) implies that

(5.29) $d_{W}(P,x^{\prime}_{Y})\leq G.$

$W\subsetneq Y$ and both $W$ and $Y$ are component domains of $\rho$. Let
$\phi_{g_{Y}}(W)$ be the footprint of $W$ on $g_{Y}$ consisting of the
vertices of $g_{Y}$ which do not overlap $W$, see $\S 4$ of [MM00] and
Definition 4.9 there. Then by Lemma 4.10 of [MM00], $\phi_{g_{Y}}(W)$ is a
sequence of $1,2$ or $3$ consecutive vertices of $g_{Y}$.

If $x^{\prime}_{Y}\in\phi_{g_{Y}}(W)$ then by the definition of foot print it
does not intersect $\partial{W}$, so $d_{Y}(\partial{W},x^{\prime}_{Y})\leq
1$. Furthermore, by (5.28), $d_{Y}(x^{\prime}_{Y},P)\leq M+\epsilon$. These
two bounds combined by the triangle inequality imply that
$d_{Y}(\partial{W},P)\leq\epsilon+M+1$. But this contradicts (5.27) because
$M+\epsilon+1<e$. So we conclude that $x^{\prime}_{Y}\not\in\phi_{g_{Y}}(W)$.

Therefore, either $x_{Y}^{\prime}>\max\phi_{g_{Y}}(W)$ or
$x^{\prime}_{Y}<\min\phi_{g_{Y}}(W)$ as vertices on the geodesic
$g_{Y}\subset\mathcal{C}(S)$. We proceed to discuss these two cases. Let $u$
be the initial vertex of the geodesic $g_{Y}$ and $v$ be its final vertex.

Case 2.2.1: $x^{\prime}_{Y}<\min\phi_{g_{Y}}(W)$.

By the definition of foot print $\partial{W}$ intersects every vertex of
$g_{Y}$ between $u$ and $x^{\prime}_{Y}$. Hence by Theorem 2.6

$d_{W}(x^{\prime}_{Y},u)\leq G.$

$j_{W}^{-}$ be the initial parameter of $J_{W}$. By condition $S3$ of slices
at the beginning of $\S 5$ of [MM00] there is $w\in\phi_{g_{Y}}(W)$ such that
$w\subset\rho(j_{W}^{-})$. Let $i\in J_{Y}$ be such that $\rho(i)\supset u$.
Then since $u\leq w$ on $g_{Y}$, Theorem 2.13 (3) (Monotonicity) implies that
$i\leq j_{W}^{-}$. Then Theorem 2.13 (4) implies that

$d_{W}(u,\mu^{-})\leq M_{2}.$

Combining the above two subsurface coefficient bounds and (5.29) by the
triangle inequality we get

(5.30) $d_{W}(P,\mu^{-})\leq M_{2}+2G\leq 2M$

The second inequality above follows from the fact that $M\geq 2G$. Because
$M_{2}\geq G$ (see Lemma 6.1 (Sigma projection) in [MM00]), $M_{1}\geq 2M_{2}$
and $M\geq M_{1}$.

Let $i\in[m,i_{W}^{-}]$. Then by the first part of Claim 5.9,
$d_{W}(\rho(i),\mu^{-})\leq 2M+3$. This bound and the bound (5.30) combined by
the triangle inequality imply that $d_{W}(\rho(i),P)\leq 4M+3$. Now since
$e>4M+3$, by the definition of $E_{W}$ we have $i\in E_{W}$ so

(5.31) $[m,i^{-}_{W}]\subseteq E_{W}.$

By (5.9) $i_{W}^{-}\in J_{W}$, so by condition $S3$ of slices in $\S 5$ of
[MM00], as before, there is $w\in\pi_{g_{Y}}(W)$ such that
$w\subset\rho(i_{W}^{-})$. Let $j\in J_{Y}$ with $\rho(j)\supset
x_{Y}^{\prime}$. $x_{Y}^{\prime}\leq w$ as vertices along $g_{Y}$. Hence by
Theorem 2.13 (3) (Monotonicity) we have that $j\leq i_{W}^{-}$. So by (5.31),
$j\in E_{W}$.

Furthermore, $d_{Y}(\rho(j),P)\leq d_{Y}(x_{Y}^{\prime},P)\leq\epsilon+M$ and
$e>\epsilon+M$ so by the definition of $E_{Y}$ any $j$ as above is in the
interval $E_{Y}$. Therefore $E_{W}\cap E_{Y}\neq\emptyset$.

Case 2.2.2: $x^{\prime}_{Y}>\max\phi_{g_{Y}}(W)$.

In this case similar to Case 2.2.1 we can first show that
$d_{W}(P,\mu^{+})\leq 2M$, which again using a similar argument implies that
$E_{W}\supseteq[i_{W}^{+},n]$. Then following the exact same lines we can
conclude that $E_{W}\cap E_{Y}\neq\emptyset$.

In summary in all of the above cases we verified that (5.8) holds. Thus as we
explained earlier the lemma follows from Helly’s Theorem in dimension one. ∎

We can now complete the proof of Theorem 5.5. For any
$P\in\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ Lemma 5.6 provides $d>0$ and
$\hat{j}\in[m,n]$ such that the inequality (5.1) $d_{X}(\rho(\hat{j}),P)\leq
d$ holds for every non-annular subsurface $X\subseteq S$. Let the threshold
constant in the distance formula (2.2) be $\max\\{M_{1},d\\}$. Let $\Delta$ be
the additive constant in the distance formula corresponding to this threshold
constant. Then

$d(P,\rho(\hat{j}))\leq\Delta.$

The above bound shows that $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is contained
in the $\Delta$ neighborhood of $|\rho|$. We earlier proved that
$|\rho|\subset\Sigma_{\epsilon}(\mu^{-},\mu^{+})$. These facts together imply
that the Hausdorff distance of $|\rho|$ and
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is bounded by $\Delta$. Note that $d$
depends only on $A$ and $\epsilon$, so $\Delta$ depends only on $A$ and
$\epsilon$.

∎

###  5.2. Stability

The following theorem is the main result of this subsection

######  Theorem 5.11.

(Stable hierarchy resolution path) Given $A>0$ there is a quantifier function
$d_{A}:\mathbb{R}^{\geq 0}\times\mathbb{R}^{\geq 1}\to\mathbb{R}^{\geq 0}$
such that any hierarchy path with $A-$narrow end points $\mu^{-}$ and
$\mu^{+}$ is $d_{A}-$stable in the pants graph.

First note that by Theorem 5.5 given an $A-$narrow pair $(\mu^{-},\mu^{+})$
the Hausdorff distance of $\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ ($\epsilon>M$)
and a hierarchy path $\rho$ between $\mu^{-}$ and $\mu^{+}$ is bounded by the
constant $\Delta$ depending only on $A$ and $\epsilon$. Then if
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is $d-$stable the hierarchy path $\rho$
is stable with quantifier function $d+\Delta$. So it suffices to prove that
$\Sigma_{\epsilon}(\mu^{-},\mu^{+})$ is stable. Our strategy to prove the
stability of the $\Sigma_{\epsilon}-$hull is to show that the projection map
$\Pi$ onto the $\Sigma_{\epsilon}-$hull ($\epsilon>F$) defined in Theorem 2.16
has the following contraction property:

######  Definition 5.12.

(Contraction property) Given $R,B\geq 0$ and $0<\eta\leq 1$ a subset
$\mathcal{Y}$ of a metric space $\mathcal{X}$ is $(R,B,\eta)-$contracting if
there is a map $\Pi:\mathcal{X}\to\mathcal{Y}$ with the following property.
For every $x,y\in\mathcal{X}$ if $d(x,\Pi x)>R$ then

$d(x,y)\leq\eta d(x,\Pi y)\Longrightarrow d(\Pi x,\Pi y)\leq B$

If a map $\Pi:\mathcal{X}\to\mathcal{Y}$ satisfies the contraction property,
the coarse Lipschitz property, and coarsely preserves $\mathcal{Y}$, then for
any $K\geq 1$ and $C\geq 0$ the standard Morse lemma argument as is in the
proof of Lemma 7.1 of [MM99], gives a $d>0$, such that a $(K,C)-$quasi-
geodesic with end points in $\mathcal{Y}$ stays in the $d-$neighborhood of
$\mathcal{Y}$. In this way we get a quantifier function $d:\mathbb{R}^{\geq
1}\times\mathbb{R}^{\geq 0}\to\mathbb{R}^{\geq 0}$, depending only on $R,B$
and $\eta$ such that $\mathcal{Y}$ is $d-$stable in $\mathcal{X}$.

We take from [Mah10] the following properties of $\delta-$hyperbolic spaces
which are not necessarily locally compact (for example the curve complex).
These properties will be used in the proof of Lemma 5.18. Let $\mathcal{X}$ be
a $\delta-$hyperbolic space which is not necessarily locally compact.

######  Proposition 5.13.

Let $\zeta$ be a geodesic in $\mathcal{X}$. Given points $x,y$ in
$\mathcal{X}$ or its Gromov boundary. Let $x^{\prime},y^{\prime}$ be closet
points to $x$ and $y$ on $\zeta$, respectively. When $x$ is at the boundary,
let $\xi_{i}\in\mathcal{X}$ be a sequence of points with $\xi_{i}\to x$ as
$i\to\infty$ and let $x^{\prime}$ be the limit of $\xi_{i}$’s. Similarly for
$y$ and $y^{\prime}$. Then

  1. (i)

A geodesic $[x,w]$ connecting $x$ to $w$ any point on $\zeta$ intersects the
$3\delta$ neighborhood of $x^{\prime}$.

  2. (ii)