# A new boundary of the mapping class group111The work was partially supported
by NSFC, No: 11771456.

Lixin $\mbox{Liu}^{1}$ and Yaozhong $\mbox{Shi}^{2,}$222Corresponding author.
1Sun Yat-sen University, School of Mathematics
510275, Guangzhou, P. R. China<EMAIL_ADDRESS>
2Sun Yat-sen University, School of Mathematics
510275, Guangzhou, P. R. China<EMAIL_ADDRESS>
###### Abstract

Based on the action of the mapping class group on the space of measured
foliations, we construct a new boundary of the mapping class group and study
the structure of this boundary. As an application, for any point in
Teichmüller space, we consider the orbit of this point under the action of the
mapping class group and describe the closure of this orbit in the Thurston
compactification and the Gardiner-Masur compactification of Teichmüller space.
We also construct some new points in the Gardiner-Masur boundary of
Teichmüller space.

Keywords: Mapping class group, Measured foliation, Teichmüller space
MSC (2010): 30F60, 32G15, 57M99

##  1 Introduction

In order to study the structure of a group $G$, it is natural to equip $G$
with a boundary. For example, considering the Cayley graph of a finite
generated group and rescaling the lengths of the edges by a summable function,
we obtain a compact completion of the graph under this new metric. The
boundary of this completion is the Floyd boundary of $G$ (see [5]). Besides, a
probability measure $\nu$ on $G$ determines a random walk on $G$. There is
also a boundary determined by this random walk, which is called the Poisson
boundary. The Poisson boundary is strongly related to the harmonic functions
corresponding to the random walk (see [10]).

Let $S$ be an oriented surface of genus $g$ with $n$ punctures. We assume that
$N=3g-3+n\geq 1$. In this paper, we study a special group: the mapping class
group $Mod(S)$ of $S$. $Mod(S)$ acts on two spaces: $\mathcal{T}(S)$ and
$\mathcal{MF}$, where $\mathcal{T}(S)$ is the Teichmüller space of $S$ and
$\mathcal{MF}$ is the space of measured foliations on $S$. Let
$\mathcal{PMF}=\mathcal{MF}-\\{0\\}/R_{+}$ be the space of projective measured
foliations. Based on the action of $Mod(S)$ on $\mathcal{T}(S)$,
$\mathcal{MF}$ and $\mathcal{PMF}$, we can study the structure of $Mod(S)$.
The most important result in the study of mapping class group may be the
Nielsen-Thurston classification theorem, which states that every $f\in Mod(S)$
is one of three special types: periodic, reducible, or pseudo-Anosov. The
structure of subgroups of $Mod(S)$ was also studied by the action of $Mod(S)$
on $\mathcal{T}(S)$ and $\mathcal{MF}$ (see [14], [9], etc.).

Different boundaries of $Mod(S)$ were studied by various people (see [13],
[11], [8], etc.). In particular, Kaimanovich and Masur (see [11]) studied the
Poisson boundary of $Mod(S)$. They proved that under some natural conditions,
the Poisson boundary of $Mod(S)$ is $\mathcal{PMF}$ equipped with a unique
measure. In order to obtain their main result, they considered the action of
$Mod(S)$ on $\mathcal{PMF}$ and analyzed the asymptotic behaviour of the
action of an infinite sequence $\\{f_{n}\\}_{n=1}^{\infty}$ on $\mathcal{PMF}$
(see Subsection 1.5 in [11]). Inspired by their idea, we study the asymptotic
behaviour of $\\{f_{n}\\}_{n=1}^{\infty}$ on $\mathcal{MF}$. In the special
case that $f_{n}=f^{n}$ for a fixed $f\in Mod(S)$, we know that
(1) when $f$ is a Dehn Twist determined by a simple closed curve $\alpha$,
$\lim_{n\rightarrow\infty}\frac{1}{n}f^{n}(F)=i(\alpha,F)\alpha$ for any
$F\in\mathcal{MF}$;
(2) when $f$ is a pseudo-Anosov map with $\lambda>1$, $f(F^{u})=\lambda
F^{u}$, $f(F^{s})=\lambda^{-1}F^{s}$ and $i(F^{u},F^{s})=1$,
$\lim_{n\rightarrow\infty}\lambda^{-n}f^{n}(F)=i(F^{s},F)F^{u}$ for any
$F\in\mathcal{MF}$.

A natural generalization of these two classical results is

######  Problem 1.1.

Is the action of $Mod(S)$ on $\mathcal{MF}$ “projectively precompact”? That
is, for any sequence $f_{n}\in Mod(S)$, are there a subsequence $f_{n_{k}}$
and a sequence of positive numbers $t_{k}$ such that
$t_{k}f_{n_{k}}:\mathcal{MF}\rightarrow\mathcal{MF}$ converges to some map
$f_{0}:\mathcal{MF}\rightarrow\mathcal{MF}$?

Note that it is necessary to take a subsequence $f_{n_{k}}$ and a positive
scalar $t_{k}$ in Problem 1.1, since without these two operations, a generic
sequence $f_{n}$ is not convergent.

We settle Problem 1.1 by embedding $Mod(S)$ into an appropriate space and
constructing a new boundary of $Mod(S)$. For this, we need some notations (see
Section 3). Let $\Omega(\mathcal{MF})$ be the set of all homogeneous
measurable functions from $\mathcal{MF}$ to $\mathcal{MF}$. Note that $R_{+}$
acts on $\Omega(\mathcal{MF})$ by multiplication. Let
$P\Omega(\mathcal{MF})=\Omega(\mathcal{MF})-\\{0\\}/R_{+}$ be the projective
space of $\Omega(\mathcal{MF})$. Endow $\Omega(\mathcal{MF})$ with the
topology of pointwise convergence and $P\Omega(\mathcal{MF})$ with the
quotient topology. Considering the action of $Mod(S)$ on $\mathcal{MF}$, there
is a natural map $I:Mod(S)\rightarrow P\Omega(\mathcal{MF})$ (see Section 3).
Up to a finite normal subgroup $ker(I)$, we can identify $Mod(S)$ with its
image $I(Mod(S))$, which is denoted by $E$ for simplicity. Thus $Mod(S)$ is
nearly embedded into $P\Omega(\mathcal{MF})$ and $P\Omega(\mathcal{MF})$ is
the appropriate space for settling Problem 1.1.

With these notations, we prove that the closure $Cl(E)$ of $E=I(Mod(S))$ in
$P\Omega(\mathcal{MF})$ is metrizable and compact (Theorem 3.1). Note that
this result answers Problem 1.1 completely: by the definition of
$P\Omega(\mathcal{MF})$, identifying $Mod(S)$ with $E=I(Mod(S))$, the
compactness of $Cl(E)$ means that for any sequence $f_{n}\in Mod(S)$, there
are a subsequence $f_{n_{k}}$ and a sequence of positive numbers $t_{k}$ such
that $t_{k}f_{n_{k}}:\mathcal{MF}\rightarrow\mathcal{MF}$ converges to a map
$f_{0}:\mathcal{MF}\rightarrow\mathcal{MF}$.

Identifying $Mod(S)$ with $E=I(Mod(S))$, $\partial E=Cl(E)-E$ is a boundary of
$Mod(S)$. For the structure of $\partial E$, we prove (see Section 4)

  * •

In $Cl(E)$, $E$ is discrete and $\partial E$ is closed (Proposition 4.1).

  * •

The operations of multiplication and inverse on $Mod(S)$ extend continuously
to $Cl(E)$ (Proposition 4.3, 4.4, 4.5). But $Cl(E)$ is not indeed a group
(Remark 4.6).

  * •

Any point $p\in\partial E$ can be represented as
$[\sum_{i=1}^{k}i(E_{i},\cdot)F_{i}]$, where $\\{F_{i}\\}\,(\\{E_{i}\\})$ are
disjoint measured foliations (Theorem 4.8).

  * •

Some special points of $\partial E$ are constructed (Proposition 4.13, 4.15,
4.16). In particular, $\partial E\neq\emptyset$.

We also consider the actions of $Mod(S)$ on the Thurston compactification
$\mathcal{T}^{Th}(S)$ and the Gardiner-Masur compactification
$\mathcal{T}^{GM}(S)$ of $\mathcal{T}(S)$. The Thurston boundary is precisely
$\mathcal{PMF}$; while the structure of the Gardiner-Masur boundary $GM$ is
complex. See [6], [15], [16] and [18] for more details on the Gardiner-Masur
boundary. $Mod(S)$ acts on $\mathcal{T}^{Th}(S)$ and $\mathcal{T}^{GM}(S)$
naturally. Considering the actions of $Mod(S)$ on $\mathcal{T}^{Th}(S)$ and
$\mathcal{T}^{GM}(S)$, we have two maps

$\Pi_{Th}:Mod(S)\times\mathcal{T}^{Th}(S)\rightarrow\mathcal{T}^{Th}(S),\,(f,p)\mapsto
f(p)$

and

$\Pi_{GM}:Mod(S)\times\mathcal{T}^{GM}(S)\rightarrow\mathcal{T}^{GM}(S),\,(f,p)\mapsto
f(p).$

If we endow $Mod(S)$ with the discrete topology, then $\Pi_{Th}$ and
$\Pi_{GM}$ are both continuous. Since $Cl(E)=E\bigcup\partial E$ is a
completion of $Mod(S)$ in some sense, we extend the domains of $\Pi_{Th}$ and
$\Pi_{GM}$ to $Cl(E)\times\mathcal{T}^{Th}(S)$ and
$Cl(E)\times\mathcal{T}^{GM}(S)$, respectively (see Theorem 5.1 and Remark
5.2).

As an application, we prove Theorem 5.4, which answers the following problem:

######  Problem 1.2.

For any $x_{0}$ in $\mathcal{T}(S)$, considering the orbit $\Gamma(x_{0})$ of
$x_{0}$ under the action of $Mod(S)$, how to describe the closure of
$\Gamma(x_{0})$ in $\mathcal{T}^{Th}(S)$ or $\mathcal{T}^{GM}(S)$?

Besides, using the new boundary $\partial E$, we construct some new points in
the Gardiner-Masur boundary of $\mathcal{T}(S)$ (see Remark 5.7).

The new boundary $\partial E$ is related to a special boundary of $Mod(S)$.
For a base point $x\in\mathcal{T}(S)$, identifying $Mod(S)$ with the orbit
$\Gamma(x)$ of $x$ by a map $Mod(S)\ni f\mapsto f(x)\in\Gamma(x)$ and then
taking boundary in $\mathcal{T}^{Th}(S)$, we get a boundary of $Mod(S)$
depending upon the base point $x$. Note that this boundary is indeed the whole
$\mathcal{PMF}$ (see Theorem 5.4). Thus we may call it the “Thurston boundary”
of $Mod(S)$ with base point $x$. And then the new boundary $\partial E$ covers
each “Thurston boundary” of $Mod(S)$ (see Remark 5.6).

It may be interesting to study the relations between the new boundary
$\partial E$ of $Mod(S)$ and some known boundaries of $Mod(S)$, such as the
Floyd boundary, the Poisson boundary, etc. We will study these relations in
coming future. Besides, Hamenstädt introduced a new boundary of $Mod(S)$ (see
Section 8 in [7]). It is also interesting to compare our new boundary with
Hamenstädt’s boundary.

We may consider a more general space than $\mathcal{MF}$, that is, the space
of geodesic currents (see [1]). Note that the space of geodesic currents
includes $\mathcal{MF}$ and $\mathcal{T}(S)$. Since the construction of the
new boundary $\partial E$ is based on the action of $Mod(S)$ on $\mathcal{MF}$
and $Mod(S)$ also acts continuously on the space of geodesic currents, it is
natural to ask the following interesting problem:

######  Problem 1.3.

If we replace the space of measured foliations by the space of geodesic
currents, does the construction of the new boundary work?

This paper is organized as follows.

In Section 2, we introduce background materials on measured foliations,
Teichmüller space and the action of the mapping class group. In Section 3, we
construct the new boundary of $Mod(S)$. In Section 4, we study the structure
of the new boundary. In Section 5, we give some applications of our new
boundary.

##  2 Preliminaries

###  2.1 Measured foliations

Let $\mathcal{S}=\mathcal{S}(S)$ be the set of isotopy classes of essential
simple closed curves on $S$. For any $\alpha,\beta$ in $\mathcal{S}$, denote
by $i(\alpha,\beta)$ the geometric intersection number between $\alpha$ and
$\beta$.

Let $R_{\geq 0}=\\{x\in R:x\geq 0\\}$ and $R_{+}=\\{x\in R:x>0\\}$. Let
$R_{\geq 0}^{\mathcal{S}}$ be the space of non-negative functions on
$\mathcal{S}$ endowed with the topology of pointwise convergence. Denote the
set of weighted simple closed curves by
$R_{+}\times\mathcal{S}=\\{t\cdot\alpha:t>0,\alpha\in\mathcal{S}\\}$. It is
known that

$i_{*}:R_{+}\times\mathcal{S}\rightarrow R_{\geq 0}^{\mathcal{S}},$
$t\cdot\alpha\mapsto t\cdot i(\alpha,\cdot)$

is injective and induces a topology on $R_{+}\times\mathcal{S}$. With this
topology, $i_{*}$ is an embedding.

The closure of $i_{*}(R_{+}\times\mathcal{S})$ in $R_{\geq 0}^{\mathcal{S}}$
is called the space of measured foliations on $S$, which is denote by
$\mathcal{MF}$. $R_{+}$ acts on $R_{\geq 0}^{\mathcal{S}}$ by multiplication.
Denote $R_{\geq 0}^{\mathcal{S}}-\\{0\\}/R_{+}$ by $PR_{\geq 0}^{\mathcal{S}}$
and $\mathcal{MF}-\\{0\\}/R_{+}$ by $\mathcal{PMF}$. $\mathcal{PMF}$ is called
the space of projective measured foliations. For $F\in\mathcal{MF}-\\{0\\}$,
denote $[F]\in\mathcal{PMF}$ to be the projective class of $F$. Note that
$\mathcal{S}$ is embedded in $PR_{\geq 0}^{\mathcal{S}}$, and the closure of
$\mathcal{S}$ in $PR_{\geq 0}^{\mathcal{S}}$ is $\mathcal{PMF}$. It is well
known that $\mathcal{MF}$ is homeomorphic to $R^{6g-6+2n}$ and $\mathcal{PMF}$
is homeomorphic to $S^{6g-7+2n}$ (see [4]).

For two weighted simple closed curves $t\alpha,s\beta\in
R_{+}\times\mathcal{S}$, define their intersection number by the homogeneous
equation $i(t\alpha,s\beta)=tsi(\alpha,\beta)$. Then the intersection number
function $i$ extends continuously to
$i:\mathcal{MF}\times\mathcal{MF}\rightarrow R_{\geq 0}$.

Any $F\in\mathcal{MF}-\\{0\\}$ is represented by a singular foliation with a
transverse measure $\mu$ in the sense that for any simple closed curve
$\alpha$,

$i(F,\alpha)=\inf_{\alpha^{{}^{\prime}}}\int_{\alpha^{{}^{\prime}}}d\mu,$

where the infimum is over all simple closed curves $\alpha^{{}^{\prime}}$
homotopic to $\alpha$.

Besides, we need the definition of ergodic decomposition of a measured
foliation. A saddle connection of a foliation is a leaf connecting two
singularities (not necessarily distinct). The critical graph of a foliation is
defined to be the union of all saddle connections. The complement of the
critical graph contains finitely many connected components. Each connected
component is either a cylinder swept out by closed leaves or a so-called
minimal component (each leaf is dense). On every minimal component $D$, there
exists a finite set of ergodic transverse measures $\mu_{1},...,\mu_{n}$ such
that any transverse measure $\mu$ on $D$ can be written as
$\mu=\sum_{j=1}^{n}f_{j}\mu_{j}$ for some non-negative coefficients
$\\{f_{j}\\}$. An indecomposable component of a measured foliation is either a
cylinder with a positive weight or a minimal component $D$ with a ergodic
measure $\mu_{j}$. A measured foliation is called indecomposable if it
contains only one indecomposable component. With these notations, any measured
foliation $F$ can be uniquely represented as

$F=\sum_{i=1}^{k}F_{i},$

where each $F_{i}$ is an indecomposable measured foliation such that
$i(F_{i},F_{j})=0$ and $[F_{i}]\neq[F_{j}]$ for $i\neq j$. We call this the
ergodic decomposition of $F$.

Finitely many simple closed curves $\alpha_{1},\alpha_{2},...,\alpha_{k}$ fill
up $S$ if for any $F\in\mathcal{MF}-\\{0\\}$,

$\sum_{i=1}^{k}i(\alpha_{i},F)>0.$

We need the following result (Lemma 6.3, [17]).

######  Lemma 2.1.

Let $\\{F_{i}:i=0,1,2,...,k\\}$ be some projectively distinct indecomposable
measured foliations such that $i(F_{i},F_{j})=0\,(i\neq j)$. Then for any
$\epsilon>0$, there exists a simple closed curve $\alpha$ such that

$i(F_{i},\alpha)<\epsilon i(F_{0},\alpha),\,i=1,2,...,k.$

###  2.2 Teichmüller space and its compactifications

Let $\mathcal{T}(S)$ be the Teichmüller space of $S$. There are two equivalent
definitions of $\mathcal{T}(S)$: the set of isotopy classes of hyperbolic
metrics on $S$ and the set of isotopy classes of conformal structures on $S$.
For two hyperbolic metric $m_{1},m_{2}$ of finite area on $S$, $m_{1}$ is
equivalent to $m_{2}$ if there exists an orientation-preserving homeomorphism
$f:S\rightarrow S$ isotopic to the identity map such that $f_{*}m_{1}=m_{2}$,
where $f_{*}m_{1}$ is the push-forward of $m_{1}$ by $f$. $\mathcal{T}(S)$ is
defined to be the set of equivalence classes of hyperbolic metrics of finite
area on $S$. For two conformal structure $\mu_{1},\mu_{2}$ on $S$, $\mu_{1}$
is equivalent to $\mu_{2}$ if there exists an orientation-preserving
homeomorphism $f:S\rightarrow S$ isotopic to the identity map such that
$f_{*}\mu_{1}=\mu_{2}$, where $f_{*}\mu_{1}$ is the push-forward of $\mu_{1}$
by $f$. $\mathcal{T}(S)$ is also defined to be the set of equivalence classes
of conformal structures on $S$. By the uniformization theorem, these two
definitions are consistent.

For the definition corresponding to hyperbolic metric, we consider the
hyperbolic length function on $\mathcal{T}(S)$. For any $x\in\mathcal{T}(S)$
and $\alpha\in\mathcal{S}$, let $l(x,\alpha)$ be the hyperbolic length of the
geodesic isotopic to $\alpha$ in the hyperbolic metric corresponding to $x$.
The hyperbolic length of a simple closed curve extends continuously to the
hyperbolic length of a measured foliation. The map

$l(\cdot,\cdot):\mathcal{T}(S)\times\mathcal{MF}\rightarrow R,$ $(x,F)\mapsto
l(x,F)$

is continuous.

Thurston constructed a compactification of Teichmüller space by the hyperbolic
lengths of simple closed curves. Define a map $\widetilde{\varphi}_{Th}$ by

$\widetilde{\varphi}_{Th}:\mathcal{T}(S)\rightarrow R_{\geq 0}^{\mathcal{S}},$
$x\mapsto(l(x,\alpha))_{\alpha\in\mathcal{S}}.$

Let $pr:R_{\geq 0}^{\mathcal{S}}-\\{0\\}\rightarrow PR_{\geq 0}^{\mathcal{S}}$
be the projective map. Then the map
$\varphi_{Th}=pr\circ\widetilde{\varphi}_{Th}$ is an embedding and the closure
of the image is compact. Moreover,
$Cl(\mathcal{T}(S))-\mathcal{T}(S)=\mathcal{PMF}$. Thus we have a
compactification of $\mathcal{T}(S)$ denoted by
$\mathcal{T}^{Th}(S)=\mathcal{T}(S)\bigcup\mathcal{PMF}$.
$\mathcal{T}^{Th}(S)$ is the Thurston compactification and $\mathcal{PMF}$ is
the Thurston boundary.

A sequence $\\{x_{n}\\}_{n=1}^{\infty}$ in $\mathcal{T}(S)$ converges to a
boundary point $[F]$ in $\mathcal{PMF}$ if and only if there exists a positive
sequence $\\{t_{n}\\}_{n=1}^{\infty}$ such that
$\lim_{n\rightarrow\infty}t_{n}=0$ and
$\lim_{n\rightarrow\infty}t_{n}l(x_{n},\alpha)=i(F,\alpha)$ for any
$\alpha\in\mathcal{S}$.

For the definition corresponding to conformal structure, we consider the
extremal length function on $\mathcal{T}(S)$. For any $x\in\mathcal{T}(S)$ and
$\alpha\in\mathcal{S}$, let $Ext(x,\alpha)$ be the extremal length of $\alpha$
in the conformal structure corresponding to $x$. The extremal length of a
simple closed curve extends continuously to the extremal length of a measured
foliation. For more details on extremal length, see [12]. The map

$Ext(\cdot,\cdot):\mathcal{T}(S)\times\mathcal{MF}\rightarrow R,$
$(x,F)\mapsto Ext(x,F)$

is continuous.

Gardiner and Masur constructed a compactification of Teichmüller space by the
extremal lengths of simple closed curves in [6]. Define a map
$\widetilde{\varphi}_{GM}$ by

$\widetilde{\varphi}_{GM}:T(S)\rightarrow R_{\geq 0}^{\mathcal{S}},$
$x\mapsto(Ext^{\frac{1}{2}}(x,\alpha))_{\alpha\in\mathcal{S}}.$

The map $\varphi_{GM}=pr\circ\widetilde{\varphi}_{GM}$ is an embedding and the
closure of the image is compact. Thus we have a compactification of
$\mathcal{T}(S)$ denoted by $\mathcal{T}^{GM}(S)=\mathcal{T}(S)\bigcup GM$.
$\mathcal{T}^{GM}(S)$ is the Gardiner-Masur compactification and $GM$ is the
Gardiner-Masur boundary.

Different from the Thurston boundary $\mathcal{PMF}$, the structure of
Gardiner-Masur boundary $GM$ is much more complex. For more details on its
structure, see [15], [16], [18].

###  2.3 The action of the mapping class group

Let $Mod(S)$ be the mapping class group of surface $S$, which is the set of
isotopy classes of orientation-preserving homeomorphisms of $S$. $Mod(S)$ acts
on $\mathcal{MF}$ and $\mathcal{T}(S)$ by push-forward. Precisely, for $f\in
Mod(S)$ and $x\in\mathcal{T}(S)$, if $m$ and $\mu$ are a hyperbolic metric and
a conformal structure in the equivalence class $x$, respectively, define
$f(x)$ to be the the equivalence class of $f_{*}m$ or $f_{*}\mu$. For a
measured foliation $(F,\nu)$, define $f(F,\nu)$ to be $(f(F),f_{\ast}\nu)$.
And its action on $\mathcal{T}(S)$ extends naturally to the Thurston
compactification and the Gardiner-Masur compactification of $\mathcal{T}(S)$.
For more details on $Mod(S)$, see [3].

In this paper, we use the following convention: for any $x\in\mathcal{T}(S)$,
$f\in Mod(S)$, $F\in\mathcal{MF}$,

$l(f(x),F)=l(x,f^{-1}(F)),\,Ext(f(x),F)=Ext(x,f^{-1}(F)).$

##  3 Construction of the new boundary

Based on the action of $Mod(S)$ on the measured foliation space
$\mathcal{MF}=\mathcal{MF}(S)$, we construct a new boundary of $Mod(S)$ in
this section.

For any $f\in Mod(S)$, $f$ acts on $\mathcal{MF}$ as a homogeneous continuous
map $f:\mathcal{MF}\rightarrow\mathcal{MF}$, which is measurable in
particular. Recall that $f$ is homogeneous if for any $F\in\mathcal{MF}$,
$k\geq 0$, $f(kF)=kf(F)$. Denote the set of all homogeneous measurable maps
from $\mathcal{MF}$ to $\mathcal{MF}$ by $\Omega(\mathcal{MF})$. We endow
$\Omega(\mathcal{MF})$ with the topology of pointwise convergence.

Since $R_{+}$ acts on $\mathcal{MF}$ by multiplication, multiplying any
$f\in\Omega(\mathcal{MF})$ by a positive number $k$, we get a homogeneous
measurable map $kf:\mathcal{MF}\rightarrow\mathcal{MF},\,F\mapsto kf(F)$. Thus
$R_{+}$ also acts on $\Omega(\mathcal{MF})$ by multiplication. Then we have
the projective space
$P\Omega(\mathcal{MF})=\Omega(\mathcal{MF})-\\{0\\}/R_{+}$, where $0$ is the
zero element in $\Omega(\mathcal{MF})$. Let
$\pi:\Omega(\mathcal{MF})-\\{0\\}\rightarrow P\Omega(\mathcal{MF})$ be the
projective map. Denote $[f]=\pi(f)$ to be the projective class of
$f\in\Omega(\mathcal{MF})$.

We endow $P\Omega(\mathcal{MF})$ with the quotient topology induced by $\pi$.
Precisely, for a sequence $\\{[f_{n}]\\}_{n=0}^{\infty}$ in
$P\Omega(\mathcal{MF})$, $\lim_{n\rightarrow\infty}[f_{n}]=[f_{0}]$ if and
only if there exists a positive sequence $\\{t_{n}\\}_{n=1}^{\infty}$ such
that $t_{n}f_{n}$ converges to $f_{0}$ in the topology of pointwise
convergence.

Sending $f\in Mod(S)$ to its action $f:\mathcal{MF}\rightarrow\mathcal{MF}$,
we have a natural map

$\widetilde{I}:Mod(S)\rightarrow\Omega(\mathcal{MF}).$

Composing it with $\pi$, we have another map

$I=\pi\circ\widetilde{I}:Mod(S)\rightarrow P\Omega(\mathcal{MF}).$

The kernel $ker(I)=\\{f\in
Mod(S):[f:\mathcal{MF}\rightarrow\mathcal{MF}]=[id_{\mathcal{MF}}]\\}$ is
finite. In fact, if $f\in ker(I)$, then there exists a positive number $k$
such that for any $F\in\mathcal{MF}$, $kf(F)=F$. Since $R_{+}\mathcal{S}$ is
dense in $\mathcal{MF}$, this is equivalent to that for any
$\alpha\in\mathcal{S}$, $f(\alpha)=k\alpha$. Since $f$ sends a simple closed
curve to a simple closed curve, we have $k=1$. Thus $f\in kerI$ is equivalent
to that $f$ fixes the isotopy class of each essential simple closed curve. By
the result in page 344 of [3], we know that when the topology type $(g,n)$ of
$S$ is $(2,0),(1,1),(1,2)$ or $(0,4)$, $ker(I)$ is a subgroup of order $2$ or
$4$; in the other cases, $ker(I)$ is trivial. So up to the finite normal
subgroup $ker(I)$ (with order $1,2$ or $4$), we can identify $Mod(S)$ with its
image $I(Mod(S))$. For simplicity, denote the image $I(Mod(S))$ by $E$.

Let $Cl(E)$ be the closure of $E$ in $P\Omega(\mathcal{MF})$. The main result
of this section is

######  Theorem 3.1.

$Cl(E)$ is metrizable and compact.

Thus $Cl(E)$ is a completion of $Mod(S)$ and $\partial E=Cl(E)-E$ is a
boundary of $Mod(S)$ in some sense.

We need some preparations to prove Theorem 3.1. In order to give a clear
description to the topology of $\mathcal{MF}$, we choose $N$ simple closed
curves $\\{\alpha_{1},\alpha_{2},...,\alpha_{N}\\}$ filling up the surface $S$
such that the map

$\Phi:\mathcal{MF}\rightarrow R^{N},$
$F\mapsto\big{(}i(\alpha_{1},F),i(\alpha_{2},F),...,i(\alpha_{N},F)\big{)}$

is an embedding (see [4]). As a result, we identify $\mathcal{MF}$ with the
image $\Phi(\mathcal{MF})$ which is endowed with the Euclidean metric on
$R^{N}$.

Let $l(\cdot)=\sum_{i=1}^{N}i(\alpha_{i},\cdot):\mathcal{MF}\rightarrow
R_{\geq 0}$ be the length function on $\mathcal{MF}$ corresponding to
$\\{\alpha_{1},\alpha_{2},...,\alpha_{N}\\}$. Since
$\\{\alpha_{1},\alpha_{2},...,\alpha_{N}\\}$ fill up the surface, $l(F)=0$ if
and only if $F=0$. Recall a result from [1]:

######  Lemma 3.2.

For any $M>0$, $\\{F\in\mathcal{MF}:l(F)\leq M\\}$ is compact in
$\mathcal{MF}$.

From Lemma 3.2, we have

######  Lemma 3.3.

$\Phi(\mathcal{MF})$ is closed in $R^{N}$.

###### Proof.

Suppose $\lim_{n\rightarrow\infty}\Phi(F_{n})=f_{0}$ for some sequence
$\\{F_{n}\\}_{n=1}^{\infty}\subseteq\mathcal{MF}$ and
$f_{0}=(a_{1},a_{2},...,a_{N})\in R^{N}$. Then
$\lim_{n\rightarrow\infty}i(\alpha_{i},F_{n})=a_{i}$ for $i=1,2,...,N$. So
$l(F_{n})=\sum_{i=1}^{N}i(\alpha_{i},F_{n})\leq M$ for some $M>0$. From Lemma
3.2, there exists a subsequence $\\{F_{n_{k}}\\}_{k=1}^{\infty}$ such that
$\lim_{k\rightarrow\infty}F_{n_{k}}=F_{0}$ for some $F_{0}\in\mathcal{MF}$.
Since $\Phi$ is continuous, we have $\Phi(F_{0})=f_{0}$. Thus
$f_{0}\in\Phi(\mathcal{MF})$, which completes the proof. ∎

Let $\Omega^{{}^{\prime}}(\mathcal{MF})\subseteq\Omega(\mathcal{MF})$ be the
set of homogeneous continuous maps from $\mathcal{MF}$ to $\mathcal{MF}$. And
let $P\Omega^{{}^{\prime}}(\mathcal{MF})\subseteq P\Omega(\mathcal{MF})$ be
the projective space of $\Omega^{{}^{\prime}}(\mathcal{MF})$. Since $Mod(S)$
acts continuously on $\mathcal{MF}$, $E\subseteq
P\Omega^{{}^{\prime}}(\mathcal{MF})$.

Now we proceed to construct a metric on $P\Omega^{{}^{\prime}}(\mathcal{MF})$.
Set $\mathcal{MF}_{1}=\\{F\in\mathcal{MF}:l(F)\leq 1\\}$. For any $[f]$ in
$P\Omega^{{}^{\prime}}(\mathcal{MF})$, we define the normalized lift of $[f]$
to $\Omega^{{}^{\prime}}(\mathcal{MF})$ by

$\widehat{f}(\cdot)=\frac{f(\cdot)}{L(f)}:\mathcal{MF}\rightarrow\mathcal{MF},$

where $L(f)=\sup_{\mathcal{MF}_{1}}l(f(\cdot))$. Note that $L(f)$ is finite
because of the compactness of $\mathcal{MF}_{1}$.

Let $d$ be the Euclidean metric on $\mathcal{MF}$ induced by $\Phi$: for any
$F,G\in\mathcal{MF}$, $d(F,G)=|\Phi(F)-\Phi(G)|$, where $|\cdot|$ is the
Euclidean norm on $R^{N}$. We define a map
$\widehat{d}:P\Omega^{{}^{\prime}}(\mathcal{MF})\times
P\Omega^{{}^{\prime}}(\mathcal{MF})\rightarrow R$ as follows: for any
$[f],[g]\in P\Omega^{{}^{\prime}}(\mathcal{MF})$,

$\widehat{d}([f],[g])=\sup_{F\in\mathcal{MF}_{1}}d(\widehat{f}(F),\widehat{g}(F)).$

Note that $\widehat{d}$ is a metric on $P\Omega^{{}^{\prime}}(\mathcal{MF})$.
In fact, the symmetry and the triangle inequality come from these two
properties of metric $d$; the positive definiteness comes from the definition
of the normalized lift $\widehat{f}$.

For the metric $\widehat{d}$, we have

######  Lemma 3.4.

For any $\\{[f_{n}]\\}_{n=0}^{\infty}\subseteq
P\Omega^{{}^{\prime}}(\mathcal{MF})$,
$\lim_{n\rightarrow\infty}\widehat{d}([f_{n}],[f_{0}])=0$ if and only if there
exists a positive sequence $\\{t_{n}\\}_{n=1}^{\infty}$ such that $t_{n}f_{n}$
converges uniformly to $f_{0}$ on any compact subset of $\mathcal{MF}$.

###### Proof.

Suppose that $\lim_{n\rightarrow\infty}\widehat{d}([f_{n}],[f_{0}])=0$. Then
from the definition of metric $\widehat{d}$, we know that
$f_{n}(\cdot)/L(f_{n})$ converges uniformly to $f_{0}(\cdot)/L(f_{0})$ on
$\mathcal{MF}_{1}$. For any compact subset $A$ of $\mathcal{MF}$, set
$l(A)=\sup_{F\in A}l(F)$. Note that for any $F\in A$,
$F/l(A)\in\mathcal{MF}_{1}$. Thus we have $f_{n}(\cdot)/l(A)L(f_{n})$
converges uniformly to $f_{0}(\cdot)/l(A)L(f_{0})$ on $A$, which also implies
that $L(f_{0})f_{n}(\cdot)/L(f_{n})$ converges uniformly to $f_{0}(\cdot)$ on
$A$.

Conversely, suppose that there exists a sequence
$\\{t_{n}\\}_{n=1}^{\infty}\subseteq R_{+}$ such that $t_{n}f_{n}$ converges
uniformly to $f_{0}$ on any compact subset of $\mathcal{MF}$. In particular,
$t_{n}f_{n}$ converges uniformly to $f_{0}$ on $\mathcal{MF}_{1}$, which
implies that $\frac{t_{n}f_{n}(\cdot)}{L(t_{n}f_{n})}$ converges uniformly to
$\frac{f_{0}(\cdot)}{L(f_{0})}$ on $\mathcal{MF}_{1}$. Thus

$\lim_{n\rightarrow\infty}\hat{d}([f_{n}],[f_{0}])=\lim_{n\rightarrow\infty}\sup_{\mathcal{MF}_{1}}d(\frac{f_{n}(\cdot)}{L(f_{n})},\frac{f_{0}(\cdot)}{L(f_{0})})=\lim_{n\rightarrow\infty}\sup_{\mathcal{MF}_{1}}d(\frac{t_{n}f_{n}(\cdot)}{L(t_{n}f_{n})},\frac{f_{0}(\cdot)}{L(f_{0})})=0.$

∎

In the metric space $(P\Omega^{{}^{\prime}}(\mathcal{MF}),\widehat{d})$, $E$
is precompact:

######  Lemma 3.5.

For any sequence $\\{[f_{n}]\\}_{n=1}^{\infty}\subseteq E$, there exists a
subsequence $\\{[f_{n_{k}}]\\}_{k=1}^{\infty}$ such that

$\lim_{k\rightarrow\infty}\widehat{d}([f_{n_{k}}],[f_{0}])=0$

for some $[f_{0}]\in P\Omega^{{}^{\prime}}(\mathcal{MF})$.

###### Proof.

From the definition of $E$, we assume that $f_{n}\in Mod(S)$ ($n=1,2,...$).
Take a point $x_{0}\in\mathcal{T}(S)$. Note that the action of $Mod(S)$ on
$\mathcal{T}(S)$ is properly discontinuous. Thus by the definition of the
Thurston compactification of $\mathcal{T}(S)$, one of the followings holds:
(1) $f_{n_{k}}\equiv f_{0}$ for some subsequence
$\\{f_{n_{k}}\\}_{k=1}^{\infty}$ and $f_{0}\in Mod(S)$;
(2) $\lim_{k\rightarrow\infty}f_{n_{k}}(x_{0})=[F_{0}]$ for some subsequence
$\\{f_{n_{k}}\\}_{k=1}^{\infty}$ and $[F_{0}]\in\mathcal{PMF}$.

For the case (1), $\widehat{d}([f_{n_{k}}],[f_{0}])\equiv 0.$

For the case (2), there exists a sequence of positive numbers
$\\{t_{k}\\}_{k=1}^{\infty}$ such that for any $F\in\mathcal{MF}$,

$\lim_{k\rightarrow\infty}t_{k}l(x_{0},f^{-1}_{n_{k}}(F))=\lim_{k\rightarrow\infty}t_{k}l(f_{n_{k}}(x_{0}),F)=i(F_{0},F).$

In particular, we have
$\lim_{k\rightarrow\infty}l(x_{0},t_{k}f^{-1}_{n_{k}}(\alpha_{i}))=i(F_{0},\alpha_{i})$
for $i=1,2,...,N$. Since $\\{\alpha_{i}\\}_{i=1}^{N}$ fill up the surface, we
have $l(F_{0})=\sum_{i=1}^{N}i(\alpha_{i},F_{0})>0$, which implies that
$m\leq\sum_{i=1}^{N}l(x_{0},t_{k}f^{-1}_{n_{k}}(\alpha_{i}))\leq M$ for some
$m,M>0$. Note that $\\{F\in\mathcal{MF}:l(x_{0},F)\leq M\\}$ is compact. Thus
passing to a subsequence again, we assume that
$\lim_{k\rightarrow\infty}t_{k}f^{-1}_{n_{k}}(\alpha_{i})=F_{i}$ for some
$F_{i}\in\mathcal{MF}$ and $F_{i_{0}}\neq 0$ for some $i_{0}$.

By the definition of $\Phi$, we have

$t_{k}\Phi\circ
f_{n_{k}}(\cdot)=\big{(}i(\alpha_{i},t_{k}f_{n_{k}}\cdot)\big{)}_{i=1}^{N}=\big{(}i(t_{k}f^{-1}_{n_{k}}(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}.$

Since $i(\cdot,\cdot):\mathcal{MF}\times\mathcal{MF}\rightarrow R_{\geq 0}$ is
continuous and
$\lim_{k\rightarrow\infty}t_{k}f^{-1}_{n_{k}}(\alpha_{i})=F_{i}$, we know that
$t_{k}\Phi\circ
f_{n_{k}}(\cdot)=\big{(}i(t_{k}f^{-1}_{n_{k}}(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}$
converges uniformly to $\big{(}i(F_{i},\cdot)\big{)}_{i=1}^{N}\neq 0$ on any
compact subset of $\mathcal{MF}$. By Lemma 3.3, for any $F\in\mathcal{MF}$,
$\big{(}i(F_{i},F)\big{)}_{i=1}^{N}=\lim_{k\rightarrow\infty}\big{(}i(\alpha_{i},t_{k}f_{n_{k}}(F))\big{)}_{i=1}^{N}\in\Phi(\mathcal{MF})$,
which implies that $f_{0}=\Phi^{-1}\big{(}i(F_{i},\cdot)\big{)}_{i=1}^{N}$ is
a homogeneous continuous map from $\mathcal{MF}$ to $\mathcal{MF}$. Since
$\Phi$ is a homeomorphism, $t_{k}f_{n_{k}}$ converges uniformly to $f_{0}$ on
any compact subset of $\mathcal{MF}$. By Lemma 3.4,
$\lim_{k\rightarrow\infty}\widehat{d}([f_{n_{k}}],[f_{0}])=0$. ∎

By Lemma 3.4 and Lemma 3.5, we prove Theorem 3.1 now.
Proof of Theorem 3.1. Firstly, we prove that $Cl(E)\subseteq
P\Omega^{{}^{\prime}}(\mathcal{MF})$. Naturally, $E\subseteq
P\Omega^{{}^{\prime}}(\mathcal{MF})$. Suppose that a sequence $[f_{n}]\in E$
converges to $[f_{0}]\in Cl(E)$ in $P\Omega(\mathcal{MF})$. By Lemma 3.5,
there exists a subsequence $\\{[f_{n_{k}}]\\}_{k=1}^{\infty}$ such that
$\lim_{k\rightarrow\infty}\widehat{d}([f_{n_{k}}],[f^{{}^{\prime}}_{0}])=0$
for some $[f^{{}^{\prime}}_{0}]\in P\Omega^{{}^{\prime}}(\mathcal{MF})$. By
Lemma 3.4, there exists a sequence of positive numbers
$\\{t_{k}\\}_{k=1}^{\infty}$ such that $t_{k}f_{n_{k}}$ converges uniformly to
$f^{{}^{\prime}}_{0}$ on any compact subset of $\mathcal{MF}$. Since the
uniform convergence on compact sets is stronger than the pointwise
convergence, $t_{k}f_{n_{k}}$ converges to $f^{{}^{\prime}}_{0}$ in the
topology of pointwise convergence. By the topology of $P\Omega(\mathcal{MF})$,
$[f_{n_{k}}]$ converges to $[f^{{}^{\prime}}_{0}]$ in $P\Omega(\mathcal{MF})$,
which implies that $[f_{0}]=[f^{{}^{\prime}}_{0}]\in
P\Omega^{{}^{\prime}}(\mathcal{MF})$. Thus $Cl(E)\subseteq
P\Omega^{{}^{\prime}}(\mathcal{MF})$.

Secondly, we prove that the topology on $Cl(E)$ is coincident with that
induced by the metric $\widehat{d}$, that is, $Cl(E)$ is metrizable. From
Lemma 3.4 and the fact that the uniform convergence on compact sets is
stronger than the pointwise convergence, we know that for any sequence
$\\{[f_{n}]\\}^{\infty}_{n=0}$ in $Cl(E)$,
$\lim_{n\rightarrow\infty}\widehat{d}([f_{n}],[f_{0}])$ implies that $[f_{n}]$
converges to $[f_{0}]$ in the topology of $Cl(E)$. For the inverse direction,
suppose that $[f_{n}]$ converges to $[f_{0}]$ in the topology of $Cl(E)$. We
wish to prove that $\lim_{n\rightarrow\infty}\widehat{d}([f_{n}],[f_{0}])=0$.
We prove this by contradiction. Suppose that there exists a subsequence
$\\{[f_{n_{k}}]\\}_{k=1}^{\infty}$ such that
$\widehat{d}([f_{n_{k}}],[f_{0}])>\varepsilon$ for some $\varepsilon>0$. Then
by Lemma 3.5, passing to a subsequence again, we can assume that
$\lim_{k\rightarrow\infty}\widehat{d}([f_{n_{k}}],[f^{{}^{\prime}}_{0}])=0$
for some $[f^{{}^{\prime}}_{0}]\in P\Omega^{{}^{\prime}}(\mathcal{MF})$, which
implies that $[f^{{}^{\prime}}_{0}]\in Cl(E)$ and $[f_{n_{k}}]$ converges to
$[f^{{}^{\prime}}_{0}]$ in the topology of $Cl(E)$. Thus
$[f^{{}^{\prime}}_{0}]=[f_{0}]$. By

$0=\lim_{k\rightarrow\infty}\widehat{d}([f_{n_{k}}],[f^{{}^{\prime}}_{0}])=\lim_{k\rightarrow\infty}\widehat{d}([f_{n_{k}}],[f_{0}])\geq\varepsilon,$

we get a contradiction.

Finally, by Lemma 3.5, as a dense subset of metric space
$(Cl(E),\widehat{d})$, $E$ is precompact. Thus $Cl(E)$ is compact.∎

By the proof of Theorem 3.1, we have two useful corollaries.

######  Corollary 3.6.

The boundary point set $\partial E=Cl(E)-E$ is included in
$P\Omega^{{}^{\prime}}(\mathcal{MF})$, that is, any boundary point $p$ can be
represented by $p=[f_{p}]$, where $f_{p}$ is a homogeneous continuous map from
$\mathcal{MF}$ to $\mathcal{MF}$.

######  Corollary 3.7.

For any sequence $\\{[f_{n}]\\}_{n=0}^{\infty}$ in $Cl(E)$, the followings are
equivalent:
(1) $\lim_{n\rightarrow\infty}[f_{n}]=[f_{0}]$;
(2) there exists a sequence of positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$
such that $t_{n}f_{n}$ converges to $f_{0}$ in the topology of pointwise
convergence;
(3) there exists a sequence of positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$
such that $t_{n}f_{n}$ converges uniformly to $f_{0}$ on any compact subset of
$\mathcal{MF}$.

Corollary 3.7 means that in $Cl(E)$, the pointwise convergence and the uniform
convergence on compact sets are equivalent, which is not true in general.

##  4 The structure of the boundary

In this section, we study the structure of the boundary $\partial E=Cl(E)-E$
in details. Recall that we endow $P\Omega(\mathcal{MF})$ with the quotient
topology from the pointwise convergence on $\Omega(\mathcal{MF})$. $Cl(E)$ is
the closure of $E$ in this topology.

######  Proposition 4.1.

In $Cl(E)$, $E$ is discrete and $\partial E$ is closed.

###### Proof.

For any $[f]$ in $E$, if $[f]$ is not an isolated point in $Cl(E)$, then there
exists a sequence $\\{[f_{n}]\\}_{n=1}^{\infty}$ in $E$ such that $f_{n}\neq
f_{m}$ for $n\neq m$ and $\lim_{n\rightarrow\infty}[f_{n}]=[f]$. So there
exists a sequence of positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$ such that
$\lim_{n\rightarrow\infty}t_{n}f_{n}=f$. Choosing a point $x_{0}$ in
$\mathcal{T}(S)$, we have

$\lim_{n\rightarrow\infty}t_{n}l(f_{n}^{-1}(x_{0}),\cdot)=\lim_{n\rightarrow\infty}l(x_{0},t_{n}f_{n}(\cdot))=l(x_{0},f(\cdot))=l(f^{-1}(x_{0}),\cdot).$

Thus $f_{n}^{-1}(x_{0})$ converges to $f^{-1}(x_{0})$ in $\mathcal{T}(S)$,
which contradicts the properly discontinuity of the action of $Mod(S)$ on
$\mathcal{T}(S)$. Thus any point of $E$ is isolated in $Cl(E)$, which implies
that $E$ is discrete in $Cl(E)$.

Since $E$ is discrete in $Cl(E)$, $E$ is open in $Cl(E)$. Thus $\partial E$ is
closed in $Cl(E)$. ∎

The operations of multiplication and inverse on $Mod(S)$ extend continuously
to $Cl(E)$. For this, we need some notations. Let $\widetilde{E}=\pi^{-1}(E)$
be the inverse image of $E$ in $\Omega(\mathcal{MF})$ and $Cl(\widetilde{E})$
be the closure of $\widetilde{E}$ in $\Omega(\mathcal{MF})$. By Corollary 3.6,
$Cl(\widetilde{E})\subseteq\Omega^{{}^{\prime}}(\mathcal{MF})$. Similar to
Corollary 3.7, we have

######  Corollary 4.2.

For any sequence $\\{f_{n}\\}_{n=0}^{\infty}$ in $Cl(\widetilde{E})-\\{0\\}$,
the followings are equivalent:
(1) $\lim_{n\rightarrow\infty}f_{n}=f_{0}$, that is, $f_{n}$ converges to
$f_{0}$ in the topology of pointwise convergence;
(2) $f_{n}$ converges uniformly to $f_{0}$ on any compact subsets of
$\mathcal{MF}$.

###### Proof.

Since the uniform convergence on compact sets is stronger than the pointwise
convergence, (2) implies (1).

For the inverse direction, suppose that $f_{n}$ converges to $f_{0}$ in the
topology of pointwise convergence. Then by Corollary 3.7, there exists a
sequence of positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$ such that
$t_{n}f_{n}$ converges uniformly to $f_{0}$ on any compact subset of
$\mathcal{MF}$. In particular, $t_{n}f_{n}$ converges to $f_{0}$ in the
topology of pointwise convergence, which implies that
$\lim_{n\rightarrow\infty}t_{n}=1$. Therefore,
$f_{n}=\frac{t_{n}f_{n}}{t_{n}}$ converges uniformly to $f_{0}$ on any compact
subsets of $\mathcal{MF}$. ∎

By Corollary 4.2, we have

######  Proposition 4.3.

The map $M:Cl(\widetilde{E})-\\{0\\}\times
Cl(\widetilde{E})-\\{0\\}\rightarrow\Omega^{{}^{\prime}}(\mathcal{MF})$
defined by $(f,g)\mapsto f\circ g$ is continuous. And for any $f,g\in
Cl(\widetilde{E})-\\{0\\}$, $f\circ g\in Cl(\widetilde{E})$.

###### Proof.

Firstly, we prove the continuity of $M$. Suppose that
$\lim_{n\rightarrow\infty}f_{n}=f$ and $\lim_{n\rightarrow\infty}g_{n}=g$ in
$Cl(\widetilde{E})-\\{0\\}$. By Corollary 4.2, $f_{n}$ and $g_{n}$ converge
uniformly to $f$ and $g$ on any compact subset of $\mathcal{MF}$,
respectively. We need to prove that $f_{n}\circ g_{n}$ converges to $f\circ g$
in the topology of pointwise convergence.

Let $d$ be the Euclidean metric on $\mathcal{MF}$ induced by $\Phi$. For any
$F\in\mathcal{MF}$, since $\lim_{n\rightarrow\infty}g_{n}(F)=g(F)$, we know
that for any $\epsilon>0$, there is $N_{1}>0$ such that for any $n>N_{1}$,

$d(f\circ g_{n}(F),f\circ g(F))<\frac{\epsilon}{2}$

and $\\{g_{n}(F)\\}_{n=1}^{\infty}\subseteq B$ for some compact neighbourhood
$B$ of $g(F)$.

Since $f_{n}$ converges uniformly to $f$ on $B$, there exists $N_{2}>0$ such
that for any $n>N_{2}$ and $F\in B$,
$d\big{(}f_{n}(F),f(F)\big{)}<\frac{\epsilon}{2}$.

Thus for any $n>\max\\{N_{1},N_{2}\\}$,

$d(f_{n}\circ g_{n}(F),f\circ g(F))\leq d(f_{n}\circ g_{n}(F),f\circ
g_{n}(F))+d(f\circ g_{n}(F),f\circ
g(F))<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon,$

which implies that $f_{n}\circ g_{n}(F)$ converges to $f\circ g(F)$. Thus $M$
is continuous.

Secondly, we prove that $f,g\in Cl(\widetilde{E})-\\{0\\}$ implies $f\circ
g\in Cl(\widetilde{E})$. Take $f_{n},g_{n}\in Mod(S)$ and $t_{n},k_{n}>0$ such
that $\lim_{n\rightarrow\infty}t_{n}f_{n}=f$ and
$\lim_{n\rightarrow\infty}k_{n}g_{n}=g$. Since $M$ is continuous,
$\lim_{n\rightarrow\infty}t_{n}k_{n}f_{n}\circ g_{n}=f\circ g$, which implies
that $f\circ g\in Cl(\widetilde{E})$. ∎

By Proposition 4.3, for any two element $[f],[g]$ in $Cl(E)$, if $f\circ g\neq
0$, we can define the product of $[f]$ and $[g]$ by $[f\circ g]\in Cl(E)$. In
particular, restricting to $Mod(S)$, it is coincident with the multiplication
operation on $Mod(S)$. Thus the multiplication operation on $Mod(S)$ extends
continuously to $Cl(E)$ except in some degenerated cases ($f\circ g=0$).

For the inverse operation, we have

######  Proposition 4.4.

For any $f$ in $Cl(\widetilde{E})$, there exists a unique element
$\overline{f}$ in $Cl(\widetilde{E})$ such that
$i\big{(}f(F),G\big{)}=i\big{(}F,\overline{f}(G)\big{)}$ for any $F,G$ in
$\mathcal{MF}$. And the map $\varphi:Cl(\widetilde{E})\rightarrow
Cl(\widetilde{E}),\,f\mapsto\overline{f}$ is a homeomorphism.

###### Proof.

Firstly, we prove the existence of $\overline{f}$. For any $f$ in
$Cl(\widetilde{E})$, if $f=Kf_{0}$ for some $K\geq 0$ and $f_{0}\in Mod(S)$,
we set $\overline{f}=Kf^{-1}_{0}$.

For other cases, we assume that $\lim_{n\rightarrow\infty}t_{n}f_{n}=f$ for
some $t_{n}>0,\,f_{n}\in Mod(S)\,(n=1,2,...)$. Then we have

$\lim_{n\rightarrow\infty}\Phi(t_{n}f_{n}^{-1}(\cdot))=\lim_{n\rightarrow\infty}\big{(}i(\alpha_{i},t_{n}f_{n}^{-1}(\cdot))\big{)}_{i=1}^{N}=\lim_{n\rightarrow\infty}\big{(}i(t_{n}f_{n}(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}=\big{(}i(f(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}.$

From Lemma 3.3, we set
$f_{0}=\Phi^{-1}(\big{(}i(f(\alpha_{i}),\cdot)\big{)}_{i=1}^{N})$ and then

$\lim_{n\rightarrow\infty}t_{n}f_{n}^{-1}=f_{0}.$

Thus for any $F,G$ in $\mathcal{MF}$, we have

$i(f(F),G)=\lim_{n\rightarrow\infty}i\big{(}t_{n}f_{n}(F),G\big{)}=\lim_{n\rightarrow\infty}i\big{(}F,t_{n}f_{n}^{-1}(G)\big{)}=i\big{(}F,f_{0}(G)\big{)}.$

So we set $\overline{f}=f_{0}$.

Secondly, we prove the uniqueness of $\overline{f}$. For any $f\in
Cl(\widetilde{E})$, suppose there are two elements $f_{1},f_{2}$ such that for
any $F,G\in\mathcal{MF}$,

$i\big{(}f(F),G\big{)}=i\big{(}F,f_{1}(G)\big{)}=i\big{(}F,f_{2}(G)\big{)}.$

Then

$\Phi\big{(}f_{1}(\cdot)\big{)}=\Phi\big{(}f_{2}(\cdot)\big{)}=\big{(}i(f(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}.$

Since $\Phi$ is an embedding, we know that $f_{1}=f_{2}$.

Now we prove that $\varphi$ is a homeomorphism. Obviously, we have
$\overline{\overline{f}}=f$ for any $f$ in $Cl(\widetilde{E})$, which implies
that $\varphi^{2}=id:Cl(\widetilde{E})\rightarrow Cl(\widetilde{E})$. Thus we
only need to prove that $\varphi$ is continuous. Suppose
$\lim_{n\rightarrow\infty}f_{n}=f_{0}$ for $\\{f_{n}\\}_{n=0}^{\infty}$ in
$Cl(\widetilde{E})$. Then we have

$\lim_{n\rightarrow\infty}\Phi(\overline{f_{n}}(\cdot))=\lim_{n\rightarrow\infty}\big{(}i(\alpha_{i},\overline{f_{n}}(\cdot))\big{)}_{i=1}^{N}=\lim_{n\rightarrow\infty}\big{(}i(f_{n}(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}$
$=\big{(}i(f_{0}(\alpha_{i}),\cdot)\big{)}_{i=1}^{N}=\big{(}i(\alpha_{i},\overline{f_{0}}(\cdot))\big{)}_{i=1}^{N}=\Phi(\overline{f_{0}}(\cdot)).$

Since $\Phi$ is an embedding,
$\lim_{n\rightarrow\infty}\overline{f_{n}}=\overline{f_{0}}$. Thus $\varphi$
is continuous. ∎

We call $\overline{f}$ defined in Proposition 4.4 the conjugate of $f$. For
any $[f]\in Cl(E)$, define the conjugate of $[f]$ by $[\overline{f}]$. In
particular, for any $f\in Mod(S)$, the conjugate of $f$ is exactly the inverse
of $f$ in $Mod(S)$. Thus the inverse operation on $Mod(S)$ extends
continuously to $Cl(E)$.

There is a natural relation between the operations of multiplication and
conjugate on $Cl(\widetilde{E})$.

######  Proposition 4.5.

For any $f,g\in Cl(\widetilde{E})$, $\overline{f\circ
g}=\overline{g}\circ\overline{f}$.

###### Proof.

From the definition of the conjugate operation, we know that for any
$F,G\in\mathcal{MF}$,

$i(f\circ
g(F),G)=i(g(F),\overline{f}(G))=i(F,\overline{g}\circ\overline{f}(G)),$

which implies that $\overline{f\circ g}=\overline{g}\circ\overline{f}$. ∎

######  Remark 4.6.

By proposition 4.3, 4.4, 4.5, the natural group structure of $Mod(S)$ extends
continuously to $Cl(E)$. But $Cl(E)$ is not a group, since there are some
degenerated cases that the multiplication is not defined and the conjugate
operation on $Cl(E)$ is not indeed an inverse operation for a group.

Now we prove a lemma:

######  Lemma 4.7.

Suppose $\lim_{n\rightarrow\infty}[f_{n}]=[f_{0}]$ for some
$\\{f_{n}\\}_{n=1}^{\infty}$ in $Mod(S)$ and $[f_{0}]\in\partial E$. Then
there exists a sequence of positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$ such
that $\lim_{n\rightarrow\infty}t_{n}=0$ and
$\lim_{n\rightarrow\infty}t_{n}f_{n}=f_{0}$.

###### Proof.

Since $\lim_{n\rightarrow\infty}[f_{n}]=[f_{0}]$, there exists a sequence of
positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$ such that
$\lim_{n\rightarrow\infty}t_{n}f_{n}=f_{0}$. Now we need to prove that
$\lim_{n\rightarrow\infty}t_{n}=0$.

Choosing a point $x_{0}$ in $\mathcal{T}(S)$, we have

$\lim_{n\rightarrow\infty}t_{n}l(f^{-1}_{n}(x_{0}),\cdot)=\lim_{n\rightarrow\infty}l(x_{0},t_{n}f_{n}(\cdot))=l(x_{0},f_{0}(\cdot)).$

From the properly discontinuity of the action of $Mod(S)$ on $\mathcal{T}(S)$,
we know that $f^{-1}_{n}(x_{0})\rightarrow\infty$ in $\mathcal{T}(S)$. By the
definition of the Thurston compactification, we know that
$\lim_{n\rightarrow\infty}t_{n}l(f^{-1}_{n}(x_{0}),\cdot)=l(x_{0},f_{0}(\cdot))=i(F,\cdot)$
for some $F$ in $\mathcal{MF}$. Then we have
$\lim_{n\rightarrow\infty}t_{n}=0$. ∎

Now we give a description of the points in $\partial E$.

######  Theorem 4.8.

For any $[f]$ in $\partial E$, we have

$f(\cdot)=\sum_{i=1}^{m}i(E_{i},\cdot)F_{i},$

where $E_{i},F_{i}$ are some measured foliations with $i(E_{i},E_{j})=0$ and
$i(F_{i},F_{j})=0$ for $i,j=1,2,...,m$.

###### Proof.

Since $[f]\in\partial E$, there exists a sequence $\\{f_{n}\\}_{n=1}^{\infty}$
in $Mod(S)$ such that $\lim_{n\rightarrow\infty}[f_{n}]=[f]$. By Lemma 4.7,
there exists a sequence of positive numbers $\\{t_{n}\\}_{n=1}^{\infty}$ such
that $\lim_{n\rightarrow\infty}t_{n}=0$ and
$\lim_{n\rightarrow\infty}t_{n}f_{n}=f$ in $\Omega(\mathcal{MF})$.

As $f$ is a map from $\mathcal{MF}$ to $\mathcal{MF}$, let
$Imf=\\{f(F):F\in\mathcal{MF}\\}\subseteq\mathcal{MF}$ be the image of $f$. We
claim that for any $F,G\in Imf$, $i(F,G)=0$. For any $F,G$ in $Imf$, there are
$F_{1},G_{1}\in\mathcal{MF}$ such that $f(F_{1})=F$ and $f(G_{1})=G$. Since
$\lim_{n\rightarrow\infty}t_{n}f_{n}=f$ and
$\lim_{n\rightarrow\infty}t_{n}=0$, we have

$i(F,G)=i(f(F_{1}),f(G_{1}))=\lim_{n\rightarrow\infty}i(t_{n}f_{n}(F_{1}),t_{n}f_{n}(G_{1}))=\lim_{n\rightarrow\infty}t_{n}^{2}i(F_{1},G_{1})=0.$

From this claim, there are $m$ pairwise disjoint indecomposable measured
foliations $\\{F_{i}\\}_{i=1}^{m}$ and some nonnegative function
$f_{i}(\cdot)$ on $\mathcal{MF}$ such that

$f(\cdot)=\sum_{i=1}^{m}f_{i}(\cdot)F_{i}.$

For $1\leq i\leq m$, from Lemma 2.1, there exists a sequence of simple closed
curves $\\{\gamma_{n}\\}_{n=1}^{\infty}$ such that

$i(\gamma_{n},F_{i})>0,\,\frac{i(\gamma_{n},F_{j})}{i(\gamma_{n},F_{i})}<\frac{1}{n}\,(j\neq
i).$

Then

$\frac{i(\gamma_{n},f(\cdot))}{i(\gamma_{n},F_{i})}=f_{i}(\cdot)+\sum_{j\neq
i}^{m}\frac{i(\gamma_{n},F_{j})}{i(\gamma_{n},F_{i})}f_{j}(\cdot)\leq
f_{i}(\cdot)+\frac{1}{n}\sum_{j\neq i}^{m}f_{j}(\cdot).$

Set $G_{n}=\frac{1}{i(\gamma_{n},F_{i})}\gamma_{n}$. Then

$\lim_{n\rightarrow\infty}i(\overline{f}(G_{n}),\cdot)=\lim_{n\rightarrow\infty}\frac{i(\gamma_{n},f(\cdot))}{i(\gamma_{n},F_{i})}=f_{i}(\cdot).$

In particular, for the filling curves $\\{\alpha_{j}\\}_{j=1}^{N}$ defined in
Section 3,

$\lim_{n\rightarrow\infty}i(\overline{f}(G_{n}),\alpha_{j})=f_{i}(\alpha_{j})\,\,(j=1,2,...,N).$

Thus there exists a constant $M>0$ such that

$l(\overline{f}(G_{n}))=\sum_{j=1}^{N}i(\overline{f}(G_{n}),\alpha_{j})\leq
M.$

From Lemma 3.2, there exists a subsequence $\\{G_{n_{k}}\\}_{k=1}^{\infty}$
such that

$\lim_{k\rightarrow\infty}\overline{f}(G_{n_{k}})=E_{i}$

for some $E_{i}$ in $\mathcal{MF}$. Then we have

$f_{i}(\cdot)=\lim_{k\rightarrow\infty}i(\overline{f}(G_{n_{k}}),\cdot)=i(E_{i},\cdot).$

For $1\leq i\leq m$, we construct a measured foliation $E_{i}$ as above. Since
for any $F,G\in Im\overline{f}$, $i(F,G)=0$, we have $i(E_{i},E_{j})=0$ for
$i\neq j$. Thus we have

$f=\sum_{i=1}^{m}i(E_{i},\cdot)F_{i},$

and $E_{i},F_{i}$ are measured foliations with $i(E_{i},E_{j})=0$ and
$i(F_{i},F_{j})=0$ for $i,j=1,2,...,m$, which completes the proof. ∎

######  Remark 4.9.

If $f(\cdot)=\sum_{i=1}^{m}i(E_{i},\cdot)F_{i}$ as in Theorem 4.8, then
$\overline{f}(\cdot)=\sum_{i=1}^{m}i(F_{i},\cdot)E_{i}$.

######  Problem 4.10.

Does the converse of Theorem 4.8 hold: for any
$\\{E_{i}\\}_{i=1}^{m},\\{F_{i}\\}_{i=1}^{m}$ in $\mathcal{MF}$ with
$i(E_{i},E_{j})=0$ and $i(F_{i},F_{j})=0$ ($i,j=1,2,...,m$),
$[\sum_{i=1}^{m}i(E_{i},\cdot)F_{i}]\in\partial E$?

Now we construct some special points in $\partial E$. Firstly, we consider the
limit of the sequence $\\{f^{n}\\}_{n=1}^{\infty}$ for some $f$ in $Mod(S)$.
We need two results (see [9]).

######  Proposition 4.11.

Let $f=T_{\alpha_{1}}^{n_{1}}\circ
T_{\alpha_{2}}^{n_{2}}\circ\cdot\cdot\cdot\circ T_{\alpha_{k}}^{n_{k}}$, where
$\alpha_{1},...,\alpha_{k}$ are pairwise disjoint simple closed curves,
$T_{\alpha_{i}}$ is the Dehn Twist of $\alpha_{i}$ and $n_{i}\in Z$
$(i=1,2,...,k)$. Then for any $F\in\mathcal{MF}$, we have

$\lim_{n\rightarrow\pm\infty}\frac{f^{n}(F)}{|n|}=\sum_{i=1}^{k}|n_{i}|i(\alpha_{i},F)\alpha_{i}.$

######  Proposition 4.12.

Let $f\in Mod(S)$ be a pseudo-Anosov element such that
$f(F^{s})=\lambda^{-1}F^{s},\,f(F^{u})=\lambda F^{u}$ with
$\lambda>1,\,F^{s},F^{u}\in\mathcal{MF}$ and $i(F^{s},F^{u})=1$. Then for any
$F\in\mathcal{MF}$, we have

$\lim_{n\rightarrow\infty}\frac{f^{n}(F)}{\lambda^{n}}=i(F^{s},F)F^{u},\,\,\lim_{n\rightarrow\infty}\frac{f^{-n}(F)}{\lambda^{n}}=i(F^{u},F)F^{s}.$

From Proposition 4.11 and Proposition 4.12, we have

######  Proposition 4.13.

(1) With the assumption of Proposition 4.11, we have

$\lim_{n\rightarrow\pm\infty}[f^{n}(\cdot)]=[\sum_{i=1}^{k}|n_{i}|i(\alpha_{i},\cdot)\alpha_{i}]\in\partial
E.$

(2) With the assumption of Proposition 4.12, we have

$\lim_{n\rightarrow\infty}[f^{n}(\cdot)]=[i(F^{s},\cdot)F^{u}]\in\partial
E\,\,\,\lim_{n\rightarrow\infty}[f^{-n}(\cdot)]=[i(F^{u},\cdot)F^{s}]\in\partial
E.$

It is well known that the action of $Mod(S)$ on $\mathcal{PMF}$ is minimal,
that is, the orbit of any element of $\mathcal{PMF}$ under the action of
$Mod(S)$ is dense in $\mathcal{PMF}$ (see [4]). We extend this result a
little:

######  Lemma 4.14.

Let $Mod^{{}^{\prime}}(S)\subseteq Mod(S)$ be the set of all mapping classes
preserving the punctures of $S$ pointwise. Then the action of
$Mod^{{}^{\prime}}(S)$ on $\mathcal{PMF}$ is minimal, that is, the orbit of
any element of $\mathcal{PMF}$ under the action of $Mod^{{}^{\prime}}(S)$ is
dense in $\mathcal{PMF}$.

###### Proof.

Since $\mathcal{S}$ is dense in $\mathcal{PMF}$, we only need to prove that
for any $\alpha,\beta\in\mathcal{S}$, $\beta\in
Cl(Mod^{{}^{\prime}}(S)(\alpha))$, where
$Cl(Mod^{{}^{\prime}}(S)(\alpha))\subseteq\mathcal{PMF}$ is the closure of the
orbit of $\alpha$ under the action of $Mod^{{}^{\prime}}(S)$.

Take $\gamma\in\mathcal{S}$ such that $i(\alpha,\gamma)\neq 0$ and
$i(\beta,\gamma)\neq 0$. Let $T_{\gamma}$ and $T_{\beta}$ be the Dehn Twist of
$\gamma$ and $\beta$, respectively. Note that $T_{\gamma}$ and $T_{\beta}$
preserve each puncture of $S$. Thus $T_{\gamma},T_{\beta}\in
Mod^{{}^{\prime}}(S)$. By Theorem 4.11,
$\lim_{n\rightarrow\infty}\frac{T_{\gamma}^{n}(\alpha)}{n}=i(\alpha,\gamma)\gamma$,
which implies that $\gamma\in Cl(Mod^{{}^{\prime}}(S)(\alpha))$. Using Theorem
4.11 again, we have
$\lim_{n\rightarrow\infty}\frac{T_{\beta}^{n}(\gamma)}{n}=i(\gamma,\beta)\beta$,
which implies that $\beta\in Cl(Mod^{{}^{\prime}}(S)(\gamma))$. Thus we have
$\beta\in Cl(Mod^{{}^{\prime}}(S)(\alpha))$. ∎

From Lemma 4.14, we have

######  Proposition 4.15.

For any $F,G\in\mathcal{MF}$, $[i(F,\cdot)G]\in\partial E$.

###### Proof.

From Lemma 4.14, for a simple closed curve $\alpha$ in $S$, there are two
sequences $\\{f_{n}\\}_{n=1}^{\infty},\\{g_{n}\\}_{n=1}^{\infty}$ in
$Mod^{{}^{\prime}}(S)$ such that
$\lim_{n\rightarrow\infty}[f_{n}(\alpha)]=[F]$ and
$\lim_{n\rightarrow\infty}[g_{n}(\alpha)]=[G]$ in $\mathcal{PMF}$. Note that
$[f_{\alpha}(\cdot)]=[i(\alpha,\cdot)\alpha]\in\partial E$ by Proposition
4.13(1). Then $\lim_{n\rightarrow\infty}[g_{n}\circ f_{\alpha}\circ
f^{-1}_{n}(\cdot)]=\lim_{n\rightarrow\infty}[i(f_{n}(\alpha),\cdot)g_{n}(\alpha)]=[i(F,\cdot)G]\in\partial
E$. ∎

We extend the result of Proposition 4.15 by operation on subsurfaces. Let
$\gamma_{1},...,\gamma_{p}$ be disjoint essential simple closed curves in $S$.
After cutting along these curves, we have some connected subsurfaces
$S_{1},...,S_{k}$. Then we have

######  Proposition 4.16.

For any $a_{i}\geq 0$ $(i=1,2,...,p)$, $F_{j},G_{j}$ in $\mathcal{MF}(S_{j})$
$(j=1,2,...,k)$,

$[\sum_{i=1}^{p}a_{i}i(\gamma_{i},\cdot)\gamma_{i}+\sum_{j=1}^{k}i(F_{j},\cdot)G_{j}]\in\partial
E.$

###### Proof.

For $j=1,2,...,k$, take a simple closed curve $\beta_{j}$ in $S_{j}$. Using
Lemma 4.14 in each subsurface $S_{j}$ (seen as a punctured surface), we find
two sequences $\\{f_{n}\\}_{n=1}^{\infty},\\{g_{n}\\}_{n=1}^{\infty}$ in
$Mod(S)$ such that $\lim_{n\rightarrow\infty}[f_{n}(\beta_{j})]=[F_{j}]$,
$\lim_{n\rightarrow\infty}[g_{n}(\beta_{j})]=[G_{j}]$ for $j=1,2,...,k$ and
$f_{n}(\gamma_{i})=\gamma_{i},g_{n}(\gamma_{i})=\gamma_{i}$ for $i=1,2,...,p$.
Thus for $j=1,2,...,k$, there are two sequences of positive numbers
$\\{t^{j}_{n}\\}_{n=1}^{\infty}$, $\\{s^{j}_{n}\\}_{n=1}^{\infty}$ such that

$\lim_{n\rightarrow\infty}t^{j}_{n}f_{n}(\beta_{j})=F_{j},\,\,\,\lim_{n\rightarrow\infty}s^{j}_{n}g_{n}(\beta_{j})=G_{j}\,(j=1,2,...,k).$

From Proposition 4.13(1) and the denseness of the set of rational numbers in
$R$, we have

$[h_{n}(\cdot)]=[\sum_{i=1}^{p}a_{i}i(\gamma_{i},\cdot)+\sum_{j=1}^{k}t^{j}_{n}s^{j}_{n}i(\beta_{j},\cdot)\beta_{j}]\in\partial
E\,(n=1,2,...).$

Thus

$[g_{n}\circ h_{n}\circ
f^{-1}_{n}(\cdot)]=[\sum_{i=1}^{p}a_{i}i(\gamma_{i},\cdot)+\sum_{j=1}^{k}i(t^{j}_{n}f_{n}(\beta_{j}),\cdot)s^{j}_{n}g_{n}(\beta_{j})]\in\partial
E\,(n=1,2,...).$

Note that

$\lim_{n\rightarrow\infty}[g_{n}\circ h_{n}\circ
f^{-1}_{n}(\cdot)]=[\sum_{i=1}^{p}a_{i}i(\gamma_{i},\cdot)\gamma_{i}+\sum_{j=1}^{k}i(F_{j},\cdot)G_{j}],$

which implies that
$[\sum_{i=1}^{p}a_{i}i(\gamma_{i},\cdot)\gamma_{i}+\sum_{j=1}^{k}i(F_{j},\cdot)G_{j}]\in\partial
E$. ∎

##  5 Some applications

Since $Mod(S)$ acts continuously on the Thurston compactification
$\mathcal{T}^{Th}(S)=\mathcal{T}(S)\bigcup\mathcal{PMF}$ and the Gardiner-
Masur compactification $\mathcal{T}^{GM}(S)=\mathcal{T}(S)\bigcup GM$ of
$\mathcal{T}(S)$, we have two maps

$\Pi_{Th}:Mod(S)\times\mathcal{T}^{Th}(S)\rightarrow\mathcal{T}^{Th}(S),\,(f,p)\mapsto
f(p)$

and

$\Pi_{GM}:Mod(S)\times\mathcal{T}^{GM}(S)\rightarrow\mathcal{T}^{GM}(S),\,(f,p)\mapsto
f(p).$

If we endow $Mod(S)$ with the discrete topology, then $\Pi_{Th}$ and
$\Pi_{GM}$ are both continuous. Since $Cl(E)=E\bigcup\partial E$ is a
completion of $Mod(S)$ in some sense, it may be natural to extend the domains
of $\Pi_{Th}$ and $\Pi_{GM}$ to $Cl(E)\times\mathcal{T}^{Th}(S)$ and
$Cl(E)\times\mathcal{T}^{GM}(S)$, respectively.

For this, we need equivalent models of $\mathcal{T}^{Th}(S)$ and
$\mathcal{T}^{GM}(S)$. From the definitions of $\mathcal{T}^{Th}(S)$ and
$\mathcal{T}^{GM}(S)$, a point in $\mathcal{T}^{Th}(S)$ and
$\mathcal{T}^{GM}(S)$ are represented by $[p_{1}:\mathcal{S}\rightarrow
R_{\geq 0}]$ and $[p_{2}:\mathcal{S}\rightarrow R_{\geq 0}]$, respectively,
where $[p_{i}]\in PR_{\geq 0}^{\mathcal{S}}$ is the projective class of
$p_{i}\in R_{\geq 0}^{\mathcal{S}}$. Since $R_{+}\times\mathcal{S}$ is dense
in $\mathcal{MF}$, $p_{1}$ and $p_{2}$ extend to homogeneous continuous
functions on $\mathcal{MF}$ (see [1] and [15]). Thus a point in
$\mathcal{T}^{Th}(S)$ or $\mathcal{T}^{GM}(S)$ can be represented by the
projective class of a homogeneous continuous function on $\mathcal{MF}$. Using
these notations, the actions of $Mod(S)$ on $\mathcal{T}^{Th}(S)$ and
$\mathcal{T}^{GM}(S)$ are defined as follows: for any $f\in Mod(S)$,
$p_{1}=[p_{1}:\mathcal{MF}\rightarrow R_{\geq 0}]\in\mathcal{T}^{Th}(S)$ and
$p_{2}=[p_{2}:\mathcal{MF}\rightarrow R_{\geq 0}]\in\mathcal{T}^{GM}(S)$,
$f(p_{1})=[p_{1}\circ f^{-1}]$ and $f(p_{2})=[p_{2}\circ f^{-1}]$. Since the
inverse operation $(\cdot)^{-1}$ on $Mod(S)$ extends to the conjugate
operation $\overline{(\cdot)}$ on $Cl(E)$, we define the extensions of
$\Pi_{Th}$ and $\Pi_{GM}$ as

######  Theorem 5.1.

Let $\Delta_{1}=\\{([f],[p])\in
Cl(E)\times\mathcal{T}^{Th}(S):p\circ\overline{f}(\cdot)\neq 0\\}$ and
$\Delta_{2}=\\{([f],[p])\in
Cl(E)\times\mathcal{T}^{GM}(S):p\circ\overline{f}(\cdot)\neq 0\\}$. The two
maps $\Psi_{Th}:\Delta_{1}\rightarrow\mathcal{T}^{Th}(S)$ and
$\Psi_{GM}:\Delta_{2}\rightarrow\mathcal{T}^{GM}(S)$ defined by
$\Psi_{Th}([f],[p])=[p\circ\overline{f}(\cdot)]$ and
$\Psi_{GM}([f],[p])=[p\circ\overline{f}(\cdot)]$, respectively, are
continuous.

###### Proof.

We only prove the continuity of $\Psi_{GM}$. The continuity of $\Psi_{Th}$ can
be proved by a similar argument.

Suppose
$\\{[p_{n}]\\}_{n=0}^{\infty}\subseteq\mathcal{T}^{GM}(S),\\{[f_{n}]\\}_{n=0}^{\infty}\subseteq
Cl(E)$ and $\lim_{n\rightarrow\infty}[p_{n}]=[p_{0}]$,
$\lim_{n\rightarrow\infty}[f_{n}]=[f_{0}]$. Up to some constants, we assume
that $\lim_{n\rightarrow\infty}f_{n}=f_{0}$ in $\Omega(\mathcal{MF})$ and
$p_{n}:\mathcal{MF}\rightarrow R_{\geq 0}$ converges uniformly to
$p_{0}:\mathcal{MF}\rightarrow R_{\geq 0}$ on any compact subsets of
$\mathcal{MF}$. By Proposition 4.4,
$\lim_{n\rightarrow\infty}\overline{f_{n}}=\overline{f_{0}}$.

Observe that for any $F$ in $\mathcal{MF}$,

$|p_{0}\circ\overline{f_{0}}(F)-p_{n}\circ\overline{f_{n}}(F)|\leq|p_{0}\circ\overline{f_{0}}(F)-p_{0}\circ\overline{f_{n}}(F)|+|p_{0}\circ\overline{f_{n}}(F)-p_{n}\circ\overline{f_{n}}(F)|.$

Since $\lim_{n\rightarrow\infty}\overline{f_{n}}(F)=\overline{f_{0}}(F)$, we
know that for any $\epsilon>0$, there exists $N_{1}>0$ such that for any
$n>N_{1}$,

$|p_{0}\circ\overline{f_{0}}(F)-p_{0}\circ\overline{f_{n}}(F)|<\frac{\epsilon}{2}$

and $\\{\overline{f_{n}}(F)\\}_{n=1}^{\infty}\subseteq M$ for some compact
subset $M$ of $\mathcal{MF}$.

Since $p_{n}(\cdot)$ converges uniformly to $p_{0}(\cdot)$ on compact set $M$,
there exists $N_{2}>0$ such that for any $n>N_{2}$,

$|p_{0}\circ\overline{f_{n}}(F)-p_{n}\circ\overline{f_{n}}(F)|<\frac{\epsilon}{2}.$

Thus for any $n>\max\\{N_{1},N_{2}\\}$,

$|p_{0}\circ\overline{f_{0}}(F)-p_{n}\circ\overline{f_{n}}(F)|<\epsilon,$

which implies that for any $F\in\mathcal{MF}$,

$\lim_{n\rightarrow\infty}p_{n}\circ\overline{f_{n}}(F)=p_{0}\circ\overline{f_{0}}(F).$

By the definition of $\mathcal{T}^{GM}(S)$,

$\Psi_{GM}([f_{0}],[p_{0}])=[p_{0}\circ\overline{f_{0}}(\cdot)]=\lim_{n\rightarrow\infty}[p_{n}\circ\overline{f_{n}}(\cdot)]=\lim_{n\rightarrow\infty}\Psi_{GM}([f_{n}],[p_{n}]),$

which completes the proof. ∎

######  Remark 5.2.

(1) For $[f]\in Cl(E)$, $[p_{1}]\in\mathcal{T}^{Th}(S)$ and
$[p_{2}]\in\mathcal{T}^{GM}(S)$, it may occur that
$p_{1}\circ\overline{f}(\cdot)=0$ and $p_{2}\circ\overline{f}(\cdot)=0$, that
is, the values of $p_{1}$, $p_{2}$ on the image of $\overline{f}$ are $0$. In
these cases, $\Psi_{Th}$ and $\Psi_{GM}$ are degenerated at $([f],[p_{1}])$
and $([f],[p_{2}])$, respectively. Thus we restrict the definitions of
$\Psi_{Th}$ and $\Psi_{GM}$ on $\Delta_{1}$ and $\Delta_{2}$, respectively.
(2) For $f_{0}\in Mod(S)$, $\Psi_{Th}([f_{0}],\cdot)$ and
$\Psi_{GM}([f_{0}],\cdot)$ are defined on the whole $\mathcal{T}^{Th}(S)$ and
$\mathcal{T}^{GM}(S)$, respectively. And $\Psi_{Th}([f_{0}],\cdot)$ and
$\Psi_{GM}([f_{0}],\cdot)$ are consistent with the actions of $f_{0}$ on
$\mathcal{T}^{Th}(S)$ and $\mathcal{T}^{GM}(S)$, respectively.
(3) For $x_{0}\in\mathcal{T}(S)$, $\Psi_{Th}(\cdot,x_{0})$ and
$\Psi_{GM}(\cdot,x_{0})$ are both defined on the whole $Cl(E)$.

By Theorem 5.1, we have

######  Corollary 5.3.

For any $x\in\mathcal{T}(S)$ and sequence $\\{f_{n}\\}_{n=1}^{\infty}\subseteq
Mod(S)$, suppose that $\lim_{n\rightarrow\infty}[f_{n}]=[f_{0}]$ in $Cl(E)$
for some $[f_{0}]\in\partial E$, then
$\lim_{n\rightarrow\infty}f_{n}(x)=[l(x,\overline{f_{0}}(\cdot)]\in\mathcal{PMF}$
in $\mathcal{T}^{Th}(S)$ and
$\lim_{n\rightarrow\infty}f_{n}(x)=[Ext^{\frac{1}{2}}(x,\overline{f_{0}}(\cdot))]\in
GM$ in $\mathcal{T}^{GM}(S)$.

By Theorem 5.1 and Corollary 5.3, we answer Problem 1.2: for any $x_{0}$ in
$\mathcal{T}(S)$, considering the orbit $\Gamma(x_{0})$ of $x_{0}$ under the
action of $Mod(S)$, how to describe the closure of $\Gamma(x_{0})$ in
$\mathcal{T}^{Th}(S)$ or $\mathcal{T}^{GM}(S)$? For this, we set

$\partial E^{Th}(x_{0})=\\{\Psi_{Th}([f],x_{0}):[f]\in\partial
E\\}\subseteq\mathcal{PMF};\,\,\partial
E^{GM}(x_{0})=\\{\Psi_{GM}([f],x_{0}):[f]\in\partial E\\}\subseteq GM.$

Then we have

######  Theorem 5.4.

In $\mathcal{T}^{Th}(S)$, the closure of $\Gamma(x_{0})$ is
$\Gamma(x_{0})\cup\partial E^{Th}(x_{0})$. In $\mathcal{T}^{GM}(S)$, the
closure of $\Gamma(x_{0})$ is $\Gamma(x_{0})\cup\partial E^{GM}(x_{0})$.
What’s more, $\partial E^{Th}(x_{0})=\mathcal{PMF}$.

###### Proof.

By Corollary 5.3, $\partial E^{Th}(x_{0})$ is included in the closure of
$\Gamma(x_{0})$ in $\mathcal{T}^{Th}(S)$. Conversely, suppose
$p\in\mathcal{T}^{Th}(S)$ is an element of the closure of $\Gamma(x_{0})$ in
$\mathcal{T}^{Th}(S)$ and $p\notin\Gamma(x_{0})$. Then there exists a sequence
$\\{f_{n}\\}_{n=1}^{\infty}\subseteq Mod(S)$ such that
$\lim_{n\rightarrow\infty}f_{n}(x_{0})=p$ in $\mathcal{T}^{Th}(S)$. Since
$Mod(S)$ acts properly discontinuously on $\mathcal{T}(S)$, we know that
$p\in\mathcal{PMF}$. By Theorem 3.1 and Proposition 4.1, there exists a
subsequence $\\{f_{n_{k}}\\}_{k=1}^{\infty}$ such that
$\lim_{k\rightarrow\infty}[f_{n_{k}}]=[f_{0}]$ in $Cl(E)$ for some
$[f_{0}]\in\partial E$. By Corollary 5.3, we have

$p=\lim_{k\rightarrow\infty}f_{n_{k}}(x_{0})=\Psi_{Th}([f_{0}],x_{0})\in\partial
E^{Th}(x_{0}).$

Thus the closure of $\Gamma(x_{0})$ in $\mathcal{T}^{Th}(S)$ is
$\Gamma(x_{0})\cup\partial E^{Th}(x_{0})$.

Using a similar argument, we know that the closure of $\Gamma(x_{0})$ in
$\mathcal{T}^{GM}(S)$ is $\Gamma(x_{0})\cup\partial E^{GM}(x_{0})$.

Now we prove $\partial E^{Th}(x_{0})=\mathcal{PMF}$. From Proposition 4.13(1),
we know that for any simple closed curve $\alpha$ in $S$,
$[i(\alpha,\cdot)\alpha]\in\partial E$. Thus
$[i(\alpha,\cdot)]=[l(x_{0},i(\alpha,\cdot)\alpha)]\in\partial E^{Th}(x_{0})$.
Since the set of simple closed curves is dense in $\mathcal{PMF}$, we have
$\partial E^{Th}(x_{0})=\mathcal{PMF}$. ∎

######  Remark 5.5.

It is well-known that the action of $Mod(S)$ on $\mathcal{PMF}$ is minimal
(see [4]). This fact also implies $\partial E^{Th}(x_{0})=\mathcal{PMF}$:
since $Mod(S)$ acts properly discontinuously on $\mathcal{T}(S)$, $\partial
E^{Th}(x_{0})\cap\mathcal{PMF}\neq\emptyset$. By the minimal action of
$Mod(S)$ on $\mathcal{PMF}$, this implies $\partial
E^{Th}(x_{0})=\mathcal{PMF}$.

######  Remark 5.6.

The new boundary $\partial E$ is related to a special boundary of $Mod(S)$.
Precisely, fixed a base point $x\in\mathcal{T}(S)$, sending $f\in Mod(S)$ to
$f(x)\in\Gamma(x)\subseteq\mathcal{T}(S)$, we can identify $Mod(S)$ with the
orbit $\Gamma(x)$ naturally. By Theorem 5.4, the boundary of $\Gamma(x)$ in
$\mathcal{T}^{Th}(S)$ is $\partial E^{Th}(x)=\mathcal{PMF}$. Thus $\partial
E^{Th}(x)=\mathcal{PMF}$ can be seen as a boundary of $Mod(S)$. As a boundary
of $Mod(S)$, $\partial E^{Th}(x)$ is homeomorphic to $\mathcal{PMF}$ but
depends upon the base point $x$ heavily. Thus we get a family of boundaries
$\\{\partial E^{Th}(x):x\in\mathcal{T}(S)\\}$ of $Mod(S)$ in which each
boundary is isomorphic to $\mathcal{PMF}$. We may call each boundary $\partial
E^{Th}(x)$ the Thurston boundary with base point $x$. By Theorem 5.1, the new
boundary $\partial E$ covers each boundary $\partial E^{Th}(x)$ in this family
by a surjective continuous map $\Psi_{x}:\partial E\rightarrow\partial
E^{Th}(x),\,\,[f]\mapsto\Psi_{Th}([f],x)$.

######  Remark 5.7.

Different from the case of Thurston compactification, $\partial E^{GM}(x_{0})$
may be not the whole boundary $GM$. From the compactness of $\partial E$, we
only know that $\partial E^{GM}(x_{0})$ is a compact subset of $GM$. And
$\partial E^{GM}(x_{0})$ contains some new points different from those known
points in $GM$. A special kind of boundary point was constructed in [2]:

$[Ext^{\frac{1}{2}}\big{(}x_{0},\sum_{i=1}^{k}n_{i}i(\alpha_{i},\cdot)\alpha_{i}\big{)}],$

where $\alpha_{i}$ are pairwise disjoint simple closed curves and $n_{i}>0$.
By Proposition 4.13(1), $\partial E^{GM}(x_{0})$ contains these points.

Let
$\mathcal{T}_{\epsilon}(S)=\\{x\in\mathcal{T}(S):\underline{l}(x)>\epsilon\\}$
be the $\epsilon-$Thick part of $\mathcal{T}(S)$, where
$\underline{l}(x)=\min_{\alpha\in\mathcal{S}}l(x,\alpha)$. The following
result characterizes the points of $\partial E^{GM}(x_{0})$.

######  Theorem 5.8.

For any $p\in GM$, $p\in\partial E^{GM}(x_{0})$ for some
$x_{0}\in\mathcal{T}(S)$ if and only if there exists a sequence
$\\{p_{n}\\}_{n=1}^{\infty}\subseteq\mathcal{T}_{\epsilon}(S)$ for some
$\epsilon>0$ such that $\lim_{n\rightarrow\infty}p_{n}=p$.

###### Proof.

Suppose that $p\in\partial E^{GM}(x_{0})$ for some $x_{0}\in\mathcal{T}(S)$.
Then $p=\lim_{n\rightarrow\infty}f_{n}(x_{0})$ for some sequence
$\\{f_{n}\\}_{n=1}^{\infty}\subseteq Mod(S)$. Note that
$\underline{l}\big{(}f_{n}(x_{0})\big{)}\equiv\underline{l}(x_{0})$. Set
$\epsilon=\frac{1}{2}\underline{l}(x_{0})$. Then
$f_{n}(x_{0})\in\mathcal{T}_{\epsilon}(S)$.

Suppose that $\\{p_{n}\\}_{n=1}^{\infty}\subseteq\mathcal{T}_{\epsilon}(S)$
for some $\epsilon>0$ and $\lim_{n\rightarrow\infty}p_{n}=p$. Then from the
Mumford’s compactness criterion, we know that after projecting to the moduli
space $\mathcal{M}(S)=\mathcal{T}(S)/Mod(S)$, the sequence
$\\{p_{n}\\}_{n=1}^{\infty}$ lies in a precompact set
$\mathcal{M}_{\epsilon}(S)=\mathcal{T}_{\epsilon}(S)/Mod(S)$, which is the
$\epsilon$-thick part of $\mathcal{M}(S)$. Thus passing to a subsequence, we
assume that there exists a sequence $\\{f_{n}\\}_{n=1}^{\infty}$ in $Mod(S)$
and $x_{0}$ in $\mathcal{T}(S)$ such that

$\lim_{n\rightarrow\infty}f_{n}(p_{n})=x_{0}.$

Set $x_{n}=f_{n}(p_{n})$. Then $p_{n}=f_{n}^{-1}(x_{n})$ and
$\lim_{n\rightarrow\infty}x_{n}=x_{0}$. By the compactness of $Cl(E)$, passing
to a subsequence again, we assume that $[f_{n}^{-1}]$ converges to some $[f]$
in $Cl(E)$. Thus by Theorem 5.1,

$p=\lim_{n\rightarrow\infty}p_{n}=\lim_{n\rightarrow\infty}f_{n}^{-1}(x_{n})=\lim_{n\rightarrow\infty}\Psi_{GM}([f^{-1}_{n}],x_{n})=\Psi_{GM}([f],x_{0})=[Ext^{\frac{1}{2}}(x_{0},\overline{f}(\cdot))].$

Since $[Ext^{\frac{1}{2}}(x_{0},\overline{f}(\cdot))]=p\in GM$, we have
$[f]\in\partial E$. So $p\in\partial E^{GM}(x_{0})$. ∎

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# Many-body instability of Coulomb interacting bilayer graphene: RG approach

Oskar Vafek  Kun Yang  National High Magnetic Field Laboratory and Department
of Physics, Florida State University, Tallahassee, Florida 32306, USA

###### Abstract

Low-energy electronic structure of (unbiased) bilayer graphene is made of two
Fermi points with quadratic dispersions, if trigonal-warping and other high
order contributions are ignored. We show that as a result of this qualitative
difference from single-layer graphene, short-range (or screened Coulomb)
interactions are marginally relevant. We use renormalization group to study
their effects on low-energy properties of the system, and show that the two
quadratic Fermi points spontaneously split into four Dirac points, at zero
temperature. This results in a nematic state that spontaneously breaks the
six-fold lattice rotation symmetry (combined with layer permutation) down to a
two-fold one, with a finite transition temperature. Critical properties of the
transition and effects of trigonal warping are also discussed.

The ability to predict the nature of the low temperature state of an
interacting quantum system is one of the main goals of condensed matter
theory. Nevertheless, despite ongoing effort, no single method has proved
universally sufficient and experimental input is essentially inevitable.

Under special circumstances, however, progress can be made. In particular, in
non-interacting systems with susceptibilities diverging as the temperature
approaches zero, the inclusion of arbitrarily small interaction can be shown
to lead to a finite, but also arbitrarily small transition temperature. The
method of choice in this case is the renormalization group (RG), which has the
virtue of unbiased determination of the leading instabilityShankar (1994).

In this paper we apply the RG method to the bilayer graphene with Bernal
stackingNovoselov et al. (2006); McCann and Fal’ko (2006); Neto et al. (2009);
Geim and MacDonald (2007). While in general, the motion of the non-interacting
electrons in such potential does not lead to diverging susceptibilities since
the energy spectrum has two sets of four Dirac points in the corners of the
Brillouin zone (due to trigonal warping)McCann and Fal’ko (2006); Neto et al.
(2009), if only nearest neighbor hopping is considered each set of four Dirac
points merges into a single degenerate point with parabolic dispersion (See
Fig. 1). As the nearest neighbor hopping amplitudes are the largest, the
latter is the natural starting point of theoretical analysisNilsson et al.
(2006); Min et al. (2008).

|
---|---

---

Figure 1: (Upper left) Honeycomb bilayer unit cell. Atoms in the lower layer
(2) are marked as empty (black) circles, atoms in the upper layer (1) are
filled (red) circles. As a starting point, only the intralayer nearest
neighbor hopping amplitudes $t$ and the interlayer hopping amplitudes
$t_{\perp}$ are considered. (Upper right) Constant energy contours of the
resulting dispersion, with minima at $K=\frac{4\pi}{\sqrt{3}3a}\hat{y}$ and
$K^{\prime}$ points and maximum at $\Gamma$ point. (Lower left) The energy
dispersion of the four bands along the vertical cut in the Brillouin zone. The
band splitting at the $K$ (and $K^{\prime}$) points is $t_{\perp}$. (Lower
right) Magnification of the dispersion (in units of $t$) near the degeneracy
point (solid black) as well as the dispersion in the nematic state (dashed
red) with $\Delta_{x}\neq 0$ (See Eq.35).

We start with the tight-binding Hamiltonian for electrons hopping on the
bilayer honeycomb lattice with Bernal stacking

$\mathcal{H}=\sum_{\langle{\bf r}{\bf r}^{\prime}\rangle}\left[t_{{\bf r}{\bf
r}^{\prime}}c_{\sigma}^{\dagger}({\bf r})c_{\sigma}({\bf
r}^{\prime})+h.c.\right]+\frac{1}{2}\sum_{{\bf r}{\bf
r}^{\prime}}\delta\hat{n}({\bf r})V({\bf r}-{\bf
r}^{\prime})\delta\hat{n}({\bf r}^{\prime}),$ (1)

where, in the nearest neighbor approximation, the (real) hopping amplitudes
$t$ connect the in-plane nearest neighbor sites belonging to different
sublattices and, for one of the sublattices, also the sites vertically above
it with amplitude $t_{\perp}$. Since there are four sites in the unit cell,
there are four bands whose dispersion for the above model comes from the
solution of the eigenvalue problem:

$\displaystyle\left[\begin{array}[]{cccc}0&d^{*}_{{\bf k}}&t_{\perp}&0\\\
d_{{\bf k}}&0&0&0\\\ t_{\perp}&0&0&d_{{\bf k}}\\\ 0&0&d^{*}_{{\bf
k}}&0\end{array}\right]\left[\begin{array}[]{c}b_{1}({\bf k})\\\ a_{1}({\bf
k})\\\ a_{2}({\bf k})\\\ b_{2}({\bf k})\end{array}\right]=E({\bf
k})\left[\begin{array}[]{c}b_{1}({\bf k})\\\ a_{1}({\bf k})\\\ a_{2}({\bf
k})\\\ b_{2}({\bf k})\end{array}\right].$ (14)

We find $E({\bf k})=\pm\left(\frac{1}{2}t_{\perp}\pm\sqrt{|d_{{\bf
k}}|^{2}+\frac{1}{4}t^{2}_{\perp}}\right)$, with $d_{{\bf
k}}=t\left[2\cos\left(\frac{\sqrt{3}}{2}k_{y}a\right)e^{-\frac{i}{2}k_{x}a}+e^{ik_{x}a}\right]$.
Two of the bands are gapped (at ${\bf K},{\bf K}^{\prime}$ by $t_{\perp}$) and
become separated from the low energy pair which touches at ${\bf k}=0$ (See
Fig.1). The resulting density of states at zero energy is therefore finite.

The repulsive interaction $V({\bf r}-{\bf r}^{\prime})$ in Eq.(1) is taken to
have a finite range $\xi$ which is however much larger than the lattice
spacing $a$. This is assumed to be the correct starting point, since the full
Coulomb interactions is screenedHwang and Sarma (2008) at low energy due to
the finite density of states. The analysis starting from the $1/|{\bf r}-{\bf
r}^{\prime}|$ interaction will be postponed to a future publication.

|  |  |  |
---|---|---|---|---
Figure 2: Diagrams appearing at 1-loop RG. The vertices are either
$\delta_{\alpha\beta}$ or $\Sigma^{\mu}_{\alpha\beta}$.

Following Nilsson et. al.Nilsson et al. (2008) we project out the gapped
bands. The resulting low energy effective (imaginary time) action (which
includes both $K$ and $K^{\prime}$ valleys) is

$\displaystyle\mathcal{S}$ $\displaystyle=$ $\displaystyle\int d\tau d^{2}{\bf
r}\left[\psi^{\dagger}\left(\frac{\partial}{\partial\tau}+\sum_{a=x,y}\Sigma^{a}d^{a}_{{\bf
p}}\psi\right)\right]$ (15) $\displaystyle+$
$\displaystyle\frac{1}{2}g_{1}\int d\tau d^{2}{\bf r}\psi^{\dagger}\psi({\bf
r},\tau)\psi^{\dagger}\psi({\bf r},\tau)$ $\displaystyle+$
$\displaystyle\frac{1}{2}g_{2}\int d\tau d^{2}{\bf
r}\psi^{\dagger}\Sigma^{z}\psi({\bf r},\tau)\psi^{\dagger}\Sigma^{z}\psi({\bf
r},\tau)$ $\displaystyle+$ $\displaystyle\frac{1}{2}g_{3}\int d\tau d^{2}{\bf
r}\sum_{a=x,y}\psi^{\dagger}\Sigma^{a}\psi({\bf
r},\tau)\psi^{\dagger}\Sigma^{a}\psi({\bf r},\tau)$

where the four component Fermi (Grassman) fields

$\displaystyle\psi({\bf r},\tau)=\int^{\Lambda}_{0}\frac{d^{2}{\bf
k}}{(2\pi)^{2}}e^{i{\bf k}\cdot{\bf r}}\left[\begin{array}[]{c}a_{1}({\bf
K}+{\bf k},\tau)\\\ b_{2}({\bf K}+{\bf k},\tau)\\\ a_{1}({\bf K}^{\prime}+{\bf
k},\tau)\\\ b_{2}({\bf K}^{\prime}+{\bf k},\tau)\end{array}\right]$ (20)

and

$\displaystyle d^{x}_{{\bf k}}$ $\displaystyle=$
$\displaystyle\frac{k^{2}_{x}-k^{2}_{y}}{2m},\;\;\;d^{y}_{{\bf
k}}=\frac{2k_{x}k_{y}}{2m},$ (21) $\displaystyle\Sigma^{x}$ $\displaystyle=$
$\displaystyle
1\sigma^{x},\;\;\Sigma^{y}=\tau^{z}\sigma^{y},\;\;\Sigma^{z}=\tau^{z}\sigma^{z}.$
(22)

The Pauli matrices $\sigma_{j}$ act on the layer indices $1$-$2$ and the
$\tau$ matrices act on the valley indices ${\bf K}$-${\bf K}^{\prime}$. The
effective mass is $m=2t_{\perp}/(9t^{2})$, and $\psi$ represents
$\frac{N}{2}-$copies of the four component pseudo-spinor. $N=4$ for spin
$1/2$, and e.g. for $s=1,\ldots N$, $\psi^{\dagger}\Sigma^{z}\psi({\bf
r},\tau)=\psi_{\alpha s}^{\dagger}\Sigma_{\alpha\beta}^{z}\psi_{\beta s}.$
Note that $\Sigma^{\prime}s$ have the same multiplication table as the Pauli
$\sigma^{\prime}s$:
$\Sigma^{\mu}\Sigma^{\nu}=1_{4}\delta_{\mu\nu}+i\epsilon_{\mu\nu\lambda}\Sigma^{\lambda}$
and are traceless, too. $\Lambda$ is a momentum cutoff which restricts the
modes to the vicinity of the ${\bf K}$-${\bf K}^{\prime}$ points and whose
order of magnitude is $\lesssim\sqrt{2mt_{\perp}}$.

The coupling constant $g_{1}=\int d^{2}{\bf r}V({\bf r})$, i.e. it is the
${\bf q}=0$ Fourier component of $V({\bf r})$. The coupling constants $g_{2}$
and $g_{3}$ are zero in the starting action, but as will be shown next, they
get generated in the momentum-shell RGShankar (1994), and therefore they are
made explicit in the original action.

From simple power-counting, the (engineering) scaling dimension of the field
$\psi$ is $L^{-1}$ and $L^{2}$ for $\tau$. This makes $g_{1}$, $g_{2}$ and
$g_{3}$ marginal (at the tree-level) and the question is how they flow upon
inclusion of the loop corrections. To answer this we note that all possible
Wick contractionsShankar (1994) of four-fermion operators correspond to the
diagrams in the Figure (2). The RG equations obtained by integrating fermion
modes within a thin shell $\Lambda$ and $\Lambda/s$ (centered at the $K$
point), and $\int^{\infty}_{-\infty}\frac{d\omega}{2\pi}$, are:

$\displaystyle\frac{dg_{1}}{d\ln s}$ $\displaystyle=$
$\displaystyle\left[-4g_{1}g_{3}\right]\frac{m}{4\pi}$ (23)
$\displaystyle\frac{dg_{2}}{d\ln s}$ $\displaystyle=$
$\displaystyle\left[-4(N-1)g^{2}_{2}+4g^{2}_{3}+4g_{1}g_{2}-12g_{2}g_{3}\right]\frac{m}{4\pi}$
(24) $\displaystyle\frac{dg_{3}}{d\ln s}$ $\displaystyle=$
$\displaystyle\left[-(g_{1}-g_{3})^{2}-(g_{2}-g_{3})^{2}-2(N+1)g^{2}_{3}\right]\frac{m}{4\pi}$
(25)

Figure 3: RG flow diagram of the ratios $g_{1}/g_{3}$ and $g_{2}/g_{3}$ for
$g_{3}<0$. While the ratio $g_{1}/g_{3}$ flows to zero (even if the starting
point is $g_{2}=g_{3}=0$ and $g_{1}\neq 0$), the ratio $g_{2}/g_{3}$ flows to
a fixed value, indicating two stable and one unstable rays with slopes
$m_{1}\approx-0.525$, $m_{3}\approx 13.98$ and $m_{2}\approx 0.545$,
respectively.

While the above equations cannot be solved in a closed form, it is possible to
fully analyze the qualitative nature of the RG flows. Such analysis is
facilitated by the observation that

$\frac{dg_{3}}{d\ln s}\leq 0$

which means that, unless $g_{1}=g_{2}=g_{3}=0$ when the equality holds,
$g_{3}$ strictly decreases under RG rescaling. We can therefore trade the
parametric dependence on $s$ of $g_{1}$ and $g_{2}$ for their dependence on
$g_{3}$ and retain the direction of the RG flow. For $g_{3}<0$ ($>0$), an
increase in $d\log s$ therefore corresponds to an increase (decrease) in
$\frac{dg_{3}}{g_{3}}$. Since the system is autonomous, we can eliminate $\log
s$ and arrive at a system

$\displaystyle\frac{dg_{1}}{dg_{3}}=f\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right)$
(26)
$\displaystyle\frac{dg_{2}}{dg_{3}}=g\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right)$
(27)

where

$\displaystyle f\left(x,y\right)$ $\displaystyle=$
$\displaystyle\frac{-4x}{-x^{2}-y^{2}-2(N+2)+2x+2y}$ (28) $\displaystyle
g\left(x,y\right)$ $\displaystyle=$
$\displaystyle\frac{-4(N-1)y^{2}+4+4xy-12y}{-x^{2}-y^{2}-2(N+2)+2x+2y}$ (29)

The system of Eqs.(26)-(27) is in turn homogeneous and can therefore be
written as

$\displaystyle
g_{3}\frac{d\frac{g_{1}}{g_{3}}}{dg_{3}}=-\frac{g_{1}}{g_{3}}+f\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right)$
(30) $\displaystyle
g_{3}\frac{d\frac{g_{2}}{g_{3}}}{dg_{3}}=-\frac{g_{2}}{g_{3}}+g\left(\frac{g_{1}}{g_{3}},\frac{g_{2}}{g_{3}}\right).$
(31)

The above system has three fixed points, all of which have $g_{1}/g_{3}=0$,
while $g_{2}/g_{3}=m_{1},m_{2},m_{3}$. As shown in the Fig.(3),
$m_{1}\approx-0.525$ and $m_{3}\approx 13.98$ are sinks, while $m_{2}\approx
0.545$ has one attractive direction and one repulsive. This means that once
$g_{3}$ gets to be negative, only $g_{2}$ and $g_{3}$ become important (their
ratio being fixed) while $g_{1}$ is too small compared to $g_{3}$. To see that
this is indeed what happens if the starting point is $g_{1}(s=1)>0$ and
$g_{2}(s=1)=g_{3}(s=1)=0$, note that the Eqs.(23-25) imply that finite $g_{1}$
generates finite and negative $g_{3}$ upon first iteration while $g_{2}$
remains zero until the second iteration. This means that we start with
$g_{1}/g_{3}\rightarrow-\infty$ and $g_{2}/g_{3}=0$ which is below the (red)
separatrix, thus the flow is into the region of attraction of $m_{1}$
(Fig.(3)).

$\psi^{\dagger}\tau^{\mu}\sigma^{\nu}\psi$ | $\nu=0$ | $\nu=x$ | $\nu=y$ | $\nu=z$
---|---|---|---|---
$\mu=0$ | $0,0,0$ | $1,-1,-2N$ | $1,-1,0$ | $2,2,-4$
$\mu=x$ | $1,-1,0$ | $0,0,0$ | $2,2,-4$ | $1,-1,0$
$\mu=y$ | $1,-1,0$ | $0,0,0$ | $2,2,-4$ | $1,-1,0$
$\mu=z$ | $0,0,0$ | $1,-1,0$ | $1,-1,-2N$ | $2,2-4N,-4$
Table 1: The susceptibility coefficients $A,B,C$ in Eq.(Many-body instability of Coulomb interacting bilayer graphene: RG approach) for different particle-hole order parameters $\psi^{\dagger}\mathcal{O}_{i}\psi$. In the physical case $N=4$. $\psi_{\alpha s}(\tau^{\mu}\sigma^{\nu})_{\alpha\beta}\psi_{\beta s^{\prime}}$ | $\nu=0$ | $\nu=x$ | $\nu=y$ | $\nu=z$
---|---|---|---|---
$\mu=0$ | $-1,-1,0$ | $-2,2,-4$ | $0,0,0$ | $-1,-1,0$
$\mu=x$ | $-2,2,-4$ | $-1,-1,0$ | $-1,-1,0$ | $0,0,0$
$\mu=y$ | $-2,2,-4$ | $-1,-1,0$ | $-1,-1,0$ | $0,0,0$
$\mu=z$ | $-1,-1,0$ | $-2,2,-4$ | $0,0,0$ | $-1,-1,0$
Table 2: The susceptibility coefficients $A^{\prime},B^{\prime},C^{\prime}$ in
Eq.(Many-body instability of Coulomb interacting bilayer graphene: RG
approach) for different particle-particle order parameters
$\psi_{\alpha\sigma}\mathcal{O}^{(i)}_{\alpha\beta}\psi_{\beta\sigma^{\prime}}$.
Figure 4: Numerical integration of the susceptibilities in Eq.(Many-body
instability of Coulomb interacting bilayer graphene: RG approach) for
$g_{1}(s=1)=0.01$ and $g_{2}(s=1)=g_{3}(s=1)=0$. The strongest divergence is
towards the nematic order. (Inset) Numerically determined nematic transition
temperature in units of cutoff $T_{\Lambda}\lesssim t_{\perp}$ as a function
of the dimensionless coupling $g_{1}\frac{m}{4\pi}$.

From Eqs.(23-25) we see for the fixed ratios $g_{1}/g_{3}=0$ and
$g_{2}/g_{3}=m_{j}$, $g_{3}$ becomes large and negative, indicating a runaway
flow. Given the flow of the coupling constants we can determine the
susceptibilities towards the formation of ordered states. In particular, we
consider coupling the fermions to external sources, which correspond to the
possible broken symmetry states. We therefore have additional terms in the
action:

$\displaystyle\Delta{\mathcal{S}}$ $\displaystyle=$
$\displaystyle-\Delta_{ph}^{\mathcal{O}_{i}}\int d\tau d^{2}{\bf
r}\psi^{\dagger}\mathcal{O}_{i}\psi({\bf r},\tau)$ (32) $\displaystyle-$
$\displaystyle\Delta^{\mathcal{O}_{i}}_{pp}\int d\tau d^{2}{\bf
r}\psi_{\alpha\sigma}\mathcal{O}^{i}_{\alpha\beta}\psi_{\beta\sigma^{\prime}}({\bf
r},\tau)$

Such terms, with infinitesimal $\Delta$’s explicitly break the symmetry and so
are relevant operators. The question of instability is answered by finding the
renormalization of the verticesChubukov (2009). The one which diverges first
determines the broken symmetry states. After a straigthforward calculation we
find that for a general particle-hole order parameter
$\mathcal{O}_{i}=\tau^{\mu}\sigma^{\nu}$ where $\mu,\nu=0,1,2,3$ and
$\tau_{0}=\sigma_{0}=1$,

$\displaystyle\Delta_{ph,ren}^{\tau^{\mu}\sigma^{\nu}}$ $\displaystyle=$
$\displaystyle\Delta_{ph}^{\tau^{\mu}\sigma^{\nu}}\left(1+\left[Ag_{1}+Bg_{2}+Cg_{3}\right]\frac{m}{4\pi}\ln
s\right)$

where the coefficients $A$, $B$, and $C$ are given in the Table 1. Similarly,
for a general particle-particle order parameter
$\psi_{\alpha\sigma}\mathcal{O}^{(i)}_{\alpha\beta}\psi_{\beta\sigma^{\prime}}$

$\displaystyle\Delta_{pp,ren}^{\tau^{\mu}\sigma^{\nu}}$ $\displaystyle=$
$\displaystyle\Delta_{pp}^{\tau^{\mu}\sigma^{\nu}}\left(1+\left[A^{\prime}g_{1}+B^{\prime}g_{2}+C^{\prime}g_{3}\right]\frac{m}{4\pi}\ln
s\right)$

where the coefficients $A^{\prime}$, $B^{\prime}$, and $C^{\prime}$ are given
in the Table 2.

The instability towards a particular order occurs at an energy scale (i.e.
temperature) at which the corresponding coefficient of the $\ln s$ in
Eqs.(Many-body instability of Coulomb interacting bilayer graphene: RG
approach-Many-body instability of Coulomb interacting bilayer graphene: RG
approach) diverges. Since $N=4$ and the fixed point value of
$g_{2}/g_{3}\approx-0.525$, with $g_{3}$ large and negative, it can be seen
from Table 1 that the instability appears in the $\Sigma^{x,y}$ channel, which
as we discuss next corresponds to a nematic order. The numerical integration
of the RG equations (23-25) starting with $g_{1}(s=1)>0$ and
$g_{2}(s=1)=g_{3}(s=1)=0$ shown in Fig.(4) indeed confirms that the
susceptibility diverges fastest in this channel. Within the continuum model
and in weak coupling, the instability is therefore towards the order
parameter, which we can parametrize by a complex field

$\Delta_{nem}({\bf r})\equiv\Delta_{x}({\bf r})+i\Delta_{y}({\bf
r})=\langle\psi^{\dagger}({\bf r})\left(\Sigma^{x}+i\Sigma^{y}\right)\psi({\bf
r})\rangle.$

To see that this is indeed a nematic order, note that at ${\bf q}=0$ (1) it is
translationally invariant and (2) even under rotations by $\pi$. In fact, as
the low energy Hamitonian is invariant under arbitrary rotations by an angle
$\alpha$, i.e. $U^{\dagger}(\alpha)\mathcal{H}U(\alpha)=\mathcal{H}$, where
$U_{\alpha}=e^{-i\alpha\hat{L}_{z}}e^{-i\alpha\Sigma^{z}},\;\;L_{z}=x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}$, we find that under a rotation by $\alpha$

$\Delta_{nem}({\bf r})\rightarrow\Delta_{nem}({\bf r})e^{2i\alpha}.$

This shows that the order parameter is even under rotations by $\pi$ and odd
under rotations by $\pi/2$, which makes it nematic. For uniform
$\Delta_{nem}({\bf r})$ the quadratic degeneracy point is split into two
(massless) Dirac points by an amount proportional to the magnitude of the
order parameter and the direction given by the nematic director.

The presence of the underlaying lattice further breaks the full rotational
symmetry of the long distance effective Hamiltonian down to hexagonal symmetry
centered on $a_{2}-b_{1}$ site, where the standard operations of $C_{6v}$ must
be accompanied by the appropriate layer permutations. The two components of
the order parameter, which give finite expectation values of, for instance,
$\Delta_{x}({\bf r})=$

$\displaystyle\left\langle a^{\dagger}_{1\sigma}({\bf
r})\left(b_{2\sigma}({\bf r}-a\hat{x})-\frac{1}{2}\sum_{s=\pm}b_{2\sigma}({\bf
r}+\frac{a}{2}\hat{x}s\frac{\sqrt{3}}{2}\hat{y})\right)+h.c.\right\rangle$
(35) $\displaystyle\mbox{and}\;\;\Delta_{y}({\bf r})=$
$\displaystyle\left\langle a^{\dagger}_{1\sigma}({\bf
r})\left(\frac{\sqrt{3}}{2}\sum_{s=\pm}sb_{2\sigma}({\bf
r}+\frac{a}{2}\hat{x}+s\frac{\sqrt{3}}{2}\hat{y})\right)+h.c.\right\rangle$
(36)

form a two dimensional representation of the hexagonal group. Note that the
nematic order parameter remains even under $\pi$-rotation followed by the
layer permutation.

From the arguments above we expect that the lattice has an important effect on
the critical nature of the phase transition, which would otherwise be of
Kosterlitz-Thousless kind. The reason is the existence of the third order
invariant $\Delta^{3}_{x}-3\Delta_{x}\Delta^{2}_{y}$. As a result the finite
temperature phase transition should be described by the effective Hamiltonian

$\displaystyle\mathcal{H}_{nem}=\sum_{\langle{\bf x}{\bf
x}^{\prime}\rangle}-J\cos[2(\theta({\bf x})-\theta({\bf
x}^{\prime}))]+h\sum_{{\bf x}}\cos[6\theta({\bf x})].$ (37)

where $\Delta_{x}({\bf x})+i\Delta_{y}({\bf x})=e^{2i\theta({\bf x})}$,
$\theta\in(0,2\pi]$ and the sum runs over the vertices of the triangular sub-
lattice spanned by $a_{1}$ sites. This corresponds to the $p=3$ case of the
two dimensional planar model studied by Jose et.al.José et al. (1977) and the
concomitant absence of the Gaussian spin-wave phase. Instead there is a
continuous transition between the low temperature phase where the director
locks into one of three values and a high temperature phase where vortices
unbind. Such transition is believed to belong to the 2D three-state Potts
model universality classNelson (2002) with exponentsWu (1982) $\nu=5/6$ and
$\eta=4/15$.

Finally, we discuss the effects of the trigonal warping which splits each of
the quadratic degeneracies into four massless Dirac points, which were ignored
up to now. If we denote the energy scale associated with such terms as
$T_{trig}$, below which the dispersion must be modified, then the transition
will still occur provided that the mean-field transition temperature $T_{c}$
estimated from the above model and plotted in the inset of Fig.(4) satisfies
$T_{c}\gg T_{trig}$. For screened Coulomb interactionsHwang and Sarma (2008)
$g_{1}\frac{m}{4\pi}\sim\mathcal{O}(1)$, leading to $T_{c}\lesssim t_{\perp}$.
Since the current estimates of $T_{trig}$ are of the same order of
magnitudeZhang et al. (2008), the ultimate test is experimental.

Acknowledgements: While this paper was in preparation, we became aware of
Ref.Sun et al. (2009) where lattices with fourfold and sixfold rotational
symmetry are constructed in either case the parabolic degeneracy points are
protected by the point group symmetry. In there, the degeneracy point maps
unto itself under time reversal, unlike our $K$ and $K^{\prime}$, and nematic
was found to be stabilized (within mean-field) only at finite coupling. This
work is supported in part by NSF grant No. DMR-0704133 (KY). Part of this work
was carried out while the authors were visiting Kavli Institute for
Theoretical Physics (KITP). The work at KITP is supported in part by NSF grant
No. PHY-0551164.

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