# Metrics on Hitchin Components from Hölder Distortion

Alexander Nolte

###### Abstract.

We observe Thurston’s asymmetric metric on Teichmüller space may be expressed
in terms of the Hölder regularity of boundary maps. We then associate
$2$-dimensional stratified loci in $\mathbb{R}\mathbb{P}^{n-1}$ to
$\text{PSL}_{n}(\mathbb{R})$ Hitchin representations with $n>3$. We prove that
measuring the relative Hölder distortion of these loci gives asymmetric
metrics on the Hitchin component $\text{Hit}_{n}(S)$ with complete and
geometrically meaningful symmetrizations. These are the first known
geometrically significant complete metrics on $\text{Hit}_{n}(S)$ for $n>3$.

Let $\rho_{a}$ and $\rho_{b}$ be discrete and faithful (Fuchsian)
representations in $\text{PSL}_{2}(\mathbb{R})$ of the fundamental group
$\Gamma$ of a closed oriented hyperbolic surface $S$. There is a unique
homeomorphism $\phi_{ab}$ of the boundary $\partial\mathbb{H}$ of the
hyperbolic plane $\mathbb{H}$ that is equivariant with respect to the actions
of $\rho_{b}$ and $\rho_{a}$. Let $\alpha_{ab}$ be the supremum of the
$\alpha$ for which $\phi_{ab}$ is $\alpha$-Hölder.

Observe $d(\rho_{a},\rho_{b})=-\log\alpha_{ab}$ is non-negative and satisfies
the triangle inequality as supremal Hölder exponents are super-multiplicative
under composition and valued in $[0,1]$.

In fact, $d$ has been seen before: it is a new expression of Thurston’s
asymmetric metric $d_{\text{Th}}$ [29] on Teichmüller space $\mathcal{T}(S)$
(Theorem A). That is, $d(\rho_{a},\rho_{b})=\log\text{L}_{ab}$, where
$\text{L}_{ab}$ is the infimal Lipschitz constant of identity-isotopic
homeomorphisms from $\mathbb{H}/\rho_{a}(\Gamma)$ to
$\mathbb{H}/\rho_{b}(\Gamma)$.

This paper centers on extensions of this idea to Hitchin components
$\text{Hit}_{n}(S)$. These are the connected components of the
$\text{PSL}_{n}(\mathbb{R})$ character variety of $\Gamma$ that contain
discrete and faithful representations of $\Gamma$ in
$\text{PSL}_{2}(\mathbb{R})$-subgroups of $\text{PSL}_{n}(\mathbb{R})$ that
act irreducibly on $\mathbb{R}^{n}$. The so-called Hitchin representations
that make up $\text{Hit}_{n}(S)$ demonstrate both remarkably similar behavior
to Fuchsian representations in $\text{PSL}_{2}(\mathbb{R})$ and rich “higher
rank” phenomena. They have seen intense interest and study in recent years
(e.g. [5] [8] [12] [13] [16] [23] [31]).

Our main results are as follows. We associate to any
$\text{PSL}_{n}(\mathbb{R})$-Hitchin representation with $n>3$ a particularly
well-behaved locus $\mathcal{B}_{\rho}$ (a torus bouquet) in
$\mathbb{R}\mathbb{P}^{n-1}$. We prove that measuring the Hölder regularity of
canonical maps between these torus bouquets gives asymmetric metrics $D_{B}$
on $\text{Hit}_{n}(S)$ with completeness properties (Theorems B–D). Notably,
the symmetrizations $D_{B}^{*}$ associated to measuring bi-Hölder regularity
are complete. These are the first complete metrics on $\text{Hit}_{n}(S)$ for
$n>3$ that are defined by a geometric construction.

Much effort has been invested constructing and studying metrics on Hitchin
components. Two prior constructions are valid for all $n$: pressure metrics
from dynamics extending the Weil-Petersson metric [5], and a dynamical
extension [6] of Thurston’s asymmetric metric.

A basic feature of the theory is that the natural notion of lengths of curves
associated to $\text{PSL}_{n}(\mathbb{R})$ Hitchin representations is in
general not valued in $\mathbb{R}^{+}$, but instead in the Weyl chamber
$\mathfrak{a}^{+}$ of $\mathfrak{sl}_{n}(\mathbb{R})$ (see §2.2.1). As a
result, both prior constructions require a choice of scalar length
measurement. These constructions also rely on an entropy invariant that
remains mysterious except in isolated cases [23]. In contrast, our
construction requires neither a choice of length measurement nor the use of
dynamical invariants such as entropy.

###  1.1. Main Results

Let $\rho_{a}$ and $\rho_{b}$ now be Hitchin representations
$\Gamma\to\text{PSL}_{n}(\mathbb{R})$. Let $\mathcal{F}(\mathbb{R}^{n})$ be
the manifold of full flags of subspaces of $\mathbb{R}^{n}$, i.e. of
($n-1$)-ples $(V_{1},...,V_{n-1})$ of nested subspaces of $\mathbb{R}^{n}$ of
dimension $1,...,n-1$. Let $\partial\Gamma$ be the Gromov boundary of
$\Gamma$, which is homeomorphic to the circle and is acted on by $\Gamma$. As
for Fuchsian representations, there are canonical equivariant maps
$\xi_{a},\xi_{b}:\partial\Gamma\to\mathcal{F}(\mathbb{R}^{n})$ that respect
the dynamics of the action of $\Gamma$ on $\partial\Gamma$ [13] [14]. For
$c\in\\{a,b\\}$ write $\xi_{c}=(\xi^{1}_{c},...,\xi^{n-1}_{c})$, expanding out
the flags.

######  Definition 1.1.

For $k=1,...,n-1$ the $k$-coupling map
$\phi_{ab}^{k}:\xi^{k}_{b}(\partial\Gamma)\to\xi^{k}_{a}(\partial\Gamma)$ is
$\xi^{k}_{a}\circ(\xi^{k}_{b})^{-1}$. Let $\alpha_{ab}^{k}$ be the supremum of
the $\alpha>0$ so $\phi_{ab}^{k}$ is $\alpha$-Hölder.

The $k$-coupling distance $d_{k}$ on ${\rm{Hit}}_{n}(S)$ is defined by
$d_{k}(\rho_{a},\rho_{b})=-\log\alpha_{ab}^{k}$.

Note that the domain and target of $\phi_{ab}^{k}$ are “flipped” relative to
the arguments of $d_{k}$.

In [6], Carvajales-Dai-Pozzetti-Wienhard extend Thurston’s asymmetric metric
to stretch metrics $d_{\text{Th}}^{\varphi}$ on $\text{Hit}_{n}(S)$. Here
$\varphi$ is a functional on $\mathfrak{a}^{+}$ that determines a length
function $\ell^{\alpha_{k}}$ on $\text{PSL}_{n}(\mathbb{R})$ (§2.2.1). We
encounter the case where $\varphi$ is a simple root $\alpha_{k}$
($k=1,...,n-1$), where
$d_{\text{Th}}^{\alpha_{k}}(\rho_{a},\rho_{b})=\log\,\sup_{\gamma\in\Gamma-\\{e\\}}[\ell^{\alpha_{k}}(\rho_{b}(\gamma))/\ell^{\alpha_{k}}(\rho_{a}(\gamma))]$
([6] [23]). Our first theorem is:

######  Theorem A. (Coupling is Stretch).

On $\mathcal{T}(S)$, Thurston’s asymmetric metric is the coupling metric:
$d_{1}=d_{{\rm{Th}}}$. On ${\rm{Hit}}_{n}(S)$, simple root stretch metrics are
$k$-coupling metrics: $d_{k}=d_{{\rm{Th}}}^{\alpha_{k}}$.

As an application of Theorem A, we obtain a simplified proof of a known
comparison for $d_{\text{Th}}$. Let $d_{\text{T}}$ be Teichmüller’s metric on
$\mathcal{T}(S)$, i.e. $2d_{\text{T}}(\rho_{a},\rho_{b})$ is the logarithm of
the infimal quasiconformal constant of an identity-isotopic homeomorphism from
$\mathbb{H}/\rho_{a}(\Gamma)$ to $\mathbb{H}/\rho_{b}(\gamma)$.

######  Corollary 1.2 ([28], [32]).

$d_{\rm{Th}}\leq 2d_{\rm{T}}$.

###### Proof.

Combine Theorem A, Ahlfors’ classical lower bound of $K^{-1}$ for the Hölder
regularity of a $K$-quasiconformal disk homeomorphism [2], and the definitions
of $d_{\text{T}}$ and $d_{\text{Th}}$. ∎

Theorem A is a corollary of Theorem 1.9 in [30]. The reformulations of
$d_{\text{Th}}$ and $d_{\text{Th}}^{\alpha_{k}}$ have not been observed
before. We give a different proof than [30] of the relevant regularity
estimate for $\mathcal{T}(S)$ in §2.1. Both [30] and our proof rely crucially
on [1]; our technique is new. The proof we present in this case is written to
not use higher-rank machinery.

####  1.1.1. Hitchin Components

Our main result concerns metrics on Hitchin components that similarly measure
Hölder regularity. In §1.2, for $\rho\in\text{Hit}_{n}(S)$ with $n>3$, we
construct a stratified $2$-dimensional set $\mathcal{B}_{\rho}$ in
$\mathbb{R}\mathbb{P}^{n-1}$. We call $\mathcal{B}_{\rho}$ the torus bouquet
of $\rho$. It is a finite union of tori that are $C^{1}$ away from a
distinguished curve. See Figure 1 (Left) for an example.

As before, for any $\rho_{a},\rho_{b}\in\text{Hit}_{n}(S)$ there is a
canonical homeomorphism
$\Phi_{ab}:\mathcal{B}_{\rho_{b}}\to\mathcal{B}_{\rho_{a}}$ conjugating
$\rho_{b}$ to $\rho_{a}$. Paralleling Definition 1.1 above, we define a
bouquet coupling metric $D_{B}$ on $\text{Hit}_{n}(S)$ (Definitions 1.3-1.4).
We call the symmetrization $D_{B}^{*}$ of $D_{B}$ associated to measuring bi-
Hölder regularity the bouquet bi-coupling metric. Both $D_{B}$ and $D_{B}^{*}$
are mapping class group invariant. Our main theorem follows from bounds on
$D_{B}$ in Theorem C below:

######  Theorem B.

For $n>3$, the bouquet bi-coupling metric $D_{B}^{*}$ is a complete metric on
${\rm{Hit}}_{n}(S)$.

The metric $D_{B}^{*}$ is the first geometrically significant complete metric
on ${\rm{Hit}}_{n}(S)$ for $n>3$. Pressure metrics are never complete. The
metrics of [6] are expected to usually be incomplete, and Dai has shown that
for $n=3$ only the simple root metrics $d_{\text{Th}}^{\alpha_{k}}$ have
complete symmetrizations [7]. Bouquet (bi-)coupling metrics have the further
appeal of being “purely geometric,” in that they are defined—without
entropy—in terms of geometric constructions.

###  1.2. Bouquet Coupling

We define the torus bouquets and associated metrics of our main construction
here. Our construction captures enough information to produce metrics, while
being clean and independent of any apparent choices. What seems essential to
our results is that torus bouquets are sufficiently rich and canonically
parametrized in terms of limit data.

We remark that implementing our perspective with the limit curve
$\xi(\partial\Gamma)\subset\mathcal{F}(\mathbb{R}^{n})$ seems much less
tractable than with torus bouquets. In particular, it is not clear that the
objects obtained for $n>3$ even separate points. See §1.2.3.

####  1.2.1. The Torus Bouquet

We begin with definitions. Let $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ for
$n>3$ be Hitchin with limit map $\xi=(\xi^{1},...,\xi^{n-1})$, let
$\mathcal{K}_{n}=\\{2,...,\lfloor n/2\rfloor\\}$, and let $k_{n}=\lfloor
n/2\rfloor$ where $\lfloor\cdot\rfloor$ is the floor function. For
$k\in\mathcal{K}_{n}$, define
$\Phi_{\rho}^{k}:\partial\Gamma^{2}\to\mathbb{R}\mathbb{P}^{n-1}$ by

(1.1) $\displaystyle\Phi_{\rho}^{k}(x,y)$
$\displaystyle=\begin{cases}\xi^{k}(x)\cap\xi^{n+1-k}(y)&x\neq y,\\\
\xi^{1}(x)&x=y,\end{cases}$

and define $T_{\rho}^{k}$ to be $\Phi_{\rho}^{k}(\partial\Gamma^{2})$. For
$n=4$ the torus $T_{\rho}^{2}$ is the boundary of the domain of discontinuity
for $\rho$ in $\mathbb{R}\mathbb{P}^{3}$ [12], and is rendered in Figure 1,
Left.

We next define the $n$-bouquet $B_{n}$ to be a gluing of $k_{n}-1$ copies
$\partial\Gamma^{2}_{l}$ of $\partial\Gamma^{2}$ ($l=2,...,k_{n})$ along their
diagonals, i.e. by the equivalence relation $\sim$ specified by

(1.2) $\displaystyle p\sim q,\qquad
p=(x,x)\in\partial\Gamma^{2}_{l},\,q=(x,x)\in\partial\Gamma^{2}_{m}\,\text{
where }x\in\partial\Gamma\text{ and }l,m\in\mathcal{K}_{n}.$

The bouquet $B_{n}$ carries a natural $\Gamma$-action.

Let $\Phi_{\rho}:B_{n}\to\mathbb{R}\mathbb{P}^{n-1}$ be the map induced by
$\\{\Phi_{\rho}^{k}\\}_{k=2,...,k_{n}}$ and define
$\mathcal{B}_{\rho}=\Phi_{\rho}(B_{n})$. Then $\Phi_{\rho}$ is a homeomorphism
onto $\mathcal{B}_{\rho}$ (§3). We call $\mathcal{B}_{\rho}$ the torus bouquet
of $\rho$. See Figure 1.

Figure 1. Left: the part of $\mathcal{B}_{\rho}$ for a
$\text{PSL}_{4}(\mathbb{R})$-Hitchin representation in a coordinate box in an
affine chart for $\mathbb{R}\mathbb{P}^{3}$. This is the “Fuchsian” example,
i.e. is preserved by an irreducible $\text{PSL}_{2}(\mathbb{R})$-subgroup of
$\text{PSL}_{4}(\mathbb{R})$. The singular curve is the image of
$\xi^{1}_{\rho}$. Right: sketch of the intersection of $\mathcal{B}_{\rho}$
with a $3$-dimensional subspace of the form $\xi^{4}_{\rho}(x)$ for a
$\text{PSL}_{8}(\mathbb{R})$-Hitchin representation $\rho$. The point where
the drawn curves intersect is $\xi^{1}_{\rho}(x)$. The term “bouquet” is
motivated by how the tori $T_{\rho}^{k}$ come together near the image of
$\xi^{1}_{\rho}.$

####  1.2.2. Bouquet Coupling Metrics

We now describe the associated construction of metrics.

######  Definition 1.3.

For ${\rm{PSL}}_{n}(\mathbb{R})$ Hitchin representations $\rho_{a}$ and
$\rho_{b}$ $(n>3)$, the bouquet coupling map
$\Phi_{ab}:\mathcal{B}_{\rho_{b}}\to\mathcal{B}_{\rho_{a}}$ is
$\Phi_{\rho_{a}}\circ\Phi_{\rho_{b}}^{-1}$. Let $\alpha_{ab}$ be the supremum
of the $\alpha>0$ so that $\Phi_{ab}$ is $\alpha$-Hölder and let $\beta_{ab}$
be the supremum of the $\alpha>0$ so that $\Phi_{ab}$ is $\alpha$-bi-Hölder.

######  Definition 1.4.

The bouquet coupling distance $D_{B}$ on ${\rm{Hit}}_{n}(S)$ ($n>3$) is
$D_{B}(\rho_{a},\rho_{b})=-\log\alpha_{ab}$. The bouquet bi-coupling distance
$D_{B}^{*}$ on ${\rm{Hit}}_{n}(S)$ is
$D_{B}^{*}(\rho_{a},\rho_{b})=-\log\beta_{ab}$.

This gives asymmetric and symmetric metrics that we bound in terms of
eigenvalue data. Notationally, for $\varphi\in\mathfrak{a}^{*}$ with
$\mathfrak{a}$ a Cartan subalgebra of $\mathfrak{sl}_{n}(\mathbb{R})$ the
$\varphi$-length $\ell^{\varphi}_{\rho}(\gamma)$ is $\varphi$ evaluated on the
Jordan projection $\lambda(\rho(\gamma))$ of $\rho(\gamma)$ to the Weyl
chamber $\mathfrak{a}^{+}$. See §2.2.1. We write $\alpha_{i}$ as the $i$-th
simple root $(i=1,...,n-1)$ and H as the Hilbert functional
$\alpha_{1}+\cdots+\alpha_{n-1}$.

######  Theorem C. (Metricity, Bounds).

For $n>3$ the bouquet coupling metric $D_{B}$ is an asymmetric metric on
${\rm{Hit}}_{n}(S)$, and $D_{B}^{*}$ is a metric on ${\rm{Hit}}_{n}(S)$. For
$\rho_{a},\rho_{b}\in{\rm{Hit}}_{n}(S)$,

(1.3)
$\displaystyle\max_{k=1,...,n-1}d_{\rm{Th}}^{\alpha_{k}}(\rho_{a},\rho_{b})\leq
D_{B}(\rho_{a},\rho_{b})\leq\max\left(\max_{k=1,...,n-1}d_{\rm{Th}}^{\alpha_{k}}(\rho_{a},\rho_{b}),\,\log\sup_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{b}}^{\rm{H}}(\gamma)}{\ell^{\alpha_{1}}_{\rho_{a}}(\gamma)}\right).$

The interest in and import of a metric on a moduli space is closely correlated
to the richness of the phenomena it measures. Torus bouquets reflect
interesting aspects of the geometry of Hitchin representations, while being
fairly tractable due to their low-dimensionality.

More specifically, in §3 we show each torus $T_{\rho}^{k}$ has two canonical
foliations by hyperconvex Frenet curves in projective spaces (see §2.2.2). The
leaves of these foliations are curious objects. In the case of $n=4$, one
foliation is by the boundaries of leaves of the properly convex foliated
projective structure associated to $\rho$ [12]. These domains exhibit have
projective-geometric properties. For instance, let $\mathfrak{C}$ be
collection of projective classes of properly convex domains in
$\mathbb{R}\mathbb{P}^{2}$. Recall $\mathfrak{C}$ is not $\text{T}_{0}$. These
domains give rise to the only known examples with more than one point of
minimal nonempty closed subsets of $\mathfrak{C}$ [21].

The bounds in Theorem C do not match because $T_{\rho}^{k}$ need not be
$C^{1}$ ($T_{\rho}^{2}$ is never $C^{1}$ for $n=4$ [20]). This breaks a
crucial aspect of known sharp regularity computations in similar settings.
Obtaining even these bounds requires new ideas. What is missing from an exact
expression for $D_{B}$ is an understanding of the structure of $\Phi_{ab}$
near $\xi^{1}_{\rho_{b}}(\partial\Gamma)$. In any matter, the following
asymmetric completeness property of $D_{B}$ implies Theorem B.

######  Theorem D.

For $n>3$ the bouquet coupling metric $D_{B}$ is forward proper, in the sense
that for any $\rho_{0}\in{\rm{Hit}}_{n}(S)$ and $R>0$ the forward ball
$\\{\rho\in{\rm{Hit}}_{n}(S)\mid D_{B}(\rho_{0},\rho)\leq R\\}$ is compact.

######  Remarks.

  1. (1)

We prove slightly more what we stated in Theorems A and C: we show the
coupling maps attain the regularity given by the lower bounds shown for Hölder
exponents. For instance, for $\rho_{a},\rho_{b}$ Fuchsian, the map $\phi_{ab}$
is shown to be $\exp(-d_{{\rm{Th}}}(\rho_{a},\rho_{b}))$-Hölder, rather than
only showing $\phi_{ab}$ is $\alpha^{\prime}$-Hölder for all
$\alpha^{\prime}<\exp(-d_{{\rm{Th}}}(\rho_{a},\rho_{b}))$.

  2. (2)

Theorem C and standard results imply $D_{B}$ is a continuous metric on
${\rm{Hit}}_{n}(S)$. The relevant facts are that limit cones of Hitchin
representations are contained in the interior of $\mathfrak{a}^{+}$ and vary
continuously ([23] Cor. 4.9, [26]), and that $d_{\rm{Th}}^{\alpha_{k}}$ is
Finsler [6].

  3. (3)

The bounds of Theorem C applied to the Fuchsian locus
$\mathscr{F}_{n}\subset\rm{Hit}_{n}(S)$ show that for each $n$ there is a
constant $C_{n}$ so that the restriction of $D_{B}$ to $\mathscr{F}_{n}$ is
$C_{n}$-multiplicatively comparable to Thurston’s asymmetric metric. So
$D_{B}$ is genuinely asymmetric.

  4. (4)

$D_{B}$ is only defined for $n>3$. The appropriate analogues for $n=2,3$ seem
to be to use $d_{\rm{Th}}$ for $n=2$ and
$d_{\rm{Th}}^{\alpha_{1}}=d_{\rm{Th}}^{\alpha_{2}}$ for $n=3$. For $n=3$, the
asymmetric metric $d_{\rm{Th}}^{\alpha_{1}}$ has the same completeness
properties as $D_{B}$ for reasons that fail in all higher ranks.

  5. (5)

Theorem A here and Theorem 1.3 in [6] show $d_{k}$ is an asymmetric metric on
${\rm{Hit}}_{n}(S)$, except in the case where $n>3$ is even and $k=n/2$.

####  1.2.3. Flag Coupling

We describe our coupling construction for limit curves in
$\mathcal{F}(\mathbb{R}^{n})$, and the technical complications the
construction faces. As this is non-central, we are brisk.

We expect that there are canonical parametrizations of larger subspaces than
the torus bouquets. The model here is the work of Guichard-Wienhard in §4.1 of
[12] for $n=4$ in $\mathbb{R}\mathbb{P}^{3}$, where canonical
parameterizations of each connected component of
$\mathbb{R}\mathbb{P}^{3}-\mathcal{B}_{\rho}$ are produced. We expect the
involved maps to quickly become complicated and involve a choice from a finite
collection of canonical maps [20].

######  Definition 1.5.

For $\rho_{a},\rho_{b}\in{\rm{Hit}}_{n}(S)$, the flag coupling map
$\phi_{ab}^{\mathcal{F}}:\xi_{b}(\partial\Gamma)\to\xi_{a}(\partial\Gamma)$ is
$\xi_{a}\circ\xi_{b}^{-1}$. Let $\alpha_{ab}^{\mathcal{F}}$ be the supremum of
the $\alpha>0$ so $\phi_{ab}^{\mathcal{F}}$ is $\alpha$-Hölder, and
$d_{\mathcal{F}}(\rho_{a},\rho_{b})=-\log\alpha_{ab}^{\mathcal{F}}$.

A similar argument to Theorems A and C above, which we omit, shows that

(1.4)
$\displaystyle\alpha_{ab}^{\mathcal{F}}=\min\left(1,\inf_{\gamma\in\Gamma-\\{e\\}}\frac{\min\limits_{k=1,...,n-1}\ell_{\rho_{a}}^{\alpha_{k}}(\gamma)}{\min\limits_{k=1,...,n-1}\ell_{\rho_{b}}^{\alpha_{k}}(\gamma)}\right).$

For $n>3$, even showing that $\alpha_{ab}^{\mathcal{F}}$ is strictly less than
$1$ if $\rho_{a},\rho_{b}\in\text{Hit}_{n}(S)$ are distinct seems challenging.
The closest results to this are in [23] and [5], in particular their
application to prove that $d_{\text{Th}}^{\alpha_{k}}$ separates points in
[6].

###  1.3. Related Work

Coupling maps for Fuchsian representations go back to Nielsen [19], and the
study of their Hölder regularity to Ahlfors [2] (see also [18]). Theorem A
consists of two bounds on Hölder regularity. The lower bound is the
difficulty. The upper bound was is due to Sorvali [28], and the equality to
Tsouvalas [30]. Papadapolous-Su have noted the relationship of Sorvali’s work
and $d_{\text{Th}}$, in case of surfaces with at least one puncture [22].

Numerous regularity computations on domains and mappings associated to Anosov
representations have been carried out previously [4] [9] [27] [30] [33]. A
crucial point in obtaining optimal results in such computations is Abels-
Margulis-Soifer’s work [1] on proximality and Benoist’s further development of
the direction [3]. This technique appears in [9] and [30].

As mentioned above, Carvajales-Dai-Pozzetti-Wienhard have produced other
extensions of Thurston’s asymmetric metric in the setting of Anosov
representations [6]. The perspectives of [6] and this work are different for
Teichmüller space. Pursuing the different perspectives leads to constructions
and metrics of different flavors. Namely, the starting point of [6] is a
formula of Thurston for $d_{\text{Th}}$ [29] while our work views
$d_{\text{Th}}$ as a measurement of the Hölder distortion of boundary maps.
Neither perspective directly generalizes Thurston’s original definition of his
metric in terms of Lipschitz distortion of hyperbolic metrics.

Carvajales-Dai-Pozzetti-Wienhard have the insight that involving the dynamics
of Anosov representations allows for a compellingly general theory to be
developed. This perspective sees aspects of Anosov representations that are
important in constructions of metrics and less clear from other viewpoints.
For instance, though we reproduce simple root stretch metrics, it is not at
all clear how to prove that they separate points without the perspective of
[6].

The main contrasts of our perspective with that of [6], and what we see as the
appeals of our viewpoint, are that our construction is more directly geometric
and uses the full data of the Hitchin condition. In particular, we use the
full hyperconvexity of Hitchin representations [11] [14]. Our construction is
necessarily much less general than [6].

Simple root metrics are distinguished among the asymmetric metrics of [6] as
not requiring an entropy normalization in their definition, as Potrie-
Sambarino proved the relevant entropy is constant [23]. It is appealing that
these distinguished metrics have a geometric realization.

###  1.4. Outline

In §2 we deduce Theorem A from work of Tsouvalas [30]. We also present a
distinct and simplified proof in the case of Thurston’s asymmetric metric in
§2.1. This is a model for later arguments, and accessibile without background
on Anosov representations.

The remainder of the paper concerns the bouquet coupling metrics. The main
objective is the upper bound of Theorem C. There are three main technical
matters in its proof to navigate in our setting, beyond the ones seen in the
study of limit curves of Hitchin representations in Grassmannians or
boundaries of divisible domains (e.g. [10] [30]).

The first difficulty is that the torus bouquet $\mathcal{B}_{\rho}$ contains
all but at most one eigenline of $\rho(\gamma)$ for $\gamma\in\Gamma$.
Techniques from quantitative proximality are essential to the estimate, and
are effective in analyzing neighborhoods of attracting fixed-points of linear
maps. Application these techniques to the parts of $\mathcal{B}_{\rho}$
containing intermediate-eigenvalue eigenlines is hopeless.

The second difficulty is that $\mathcal{B}_{\rho}$ is not globally $C^{1}$ in
any sense. In the irregular region, near $\xi^{1}(\partial\Gamma)$, this makes
estimation of distances between points problematic.

The third difficulty is that the natural space that $\Gamma$ acts cocompactly
on, the space $\partial\Gamma^{(3)+}$ of positively oriented triples in
$\partial\Gamma$, is of a lower dimension than
$\mathcal{B}_{\rho}\times\mathcal{B}_{\rho}$. So one application of the
uniform convergence group property of $\partial\Gamma$ does not move an
arbitrary pair of points in $\mathcal{B}_{\rho}$ to compact family of pairs,
and further analysis is needed.

What makes these difficulties tractable is that on compact subsets of
$\mathcal{B}_{\rho}-\xi^{1}(\partial\Gamma)$ each torus $T_{\rho}^{k}$ is
$C^{1}$ and has an approximate-product structure compatible with the coupling
maps $\Phi_{ab}$ (§3). This reduces the regularity of the coupling maps on
compact subsets of $\mathcal{B}_{\rho}-\xi^{1}(\partial\Gamma)$ to the
regularity of the $k$-coupling maps $\phi_{ab}^{k}$ between limit curves in
$k$-Grassmannians. This circumvents the first challenge and reduces analysis
to a small neighborhood of $\xi^{1}(\partial\Gamma)$.

So we are left to consider the regularity of coupling maps on a neighborhood
of $\xi^{1}(\partial\Gamma)$ of a fixed small scale of our choosing. With our
freedom to set this scale we arrange for all points in $\partial\Gamma$ needed
to specify a pair of points in $\mathcal{B}_{\rho}$ to be close together, and
for numerous other technical properties to hold. This particular structure
allows for the third challenge to be addressed with an adapted application of
the uniform convergence group property of $\Gamma$.

This technical perspective of this work has novelties. In particular, our use
of control over scales under consideration to improve quality constants does
not appear in prior regularity computations discussed above. Additionally, our
use of the uniform convergence group property of $\Gamma$ in terms of the
action of $\Gamma$ on triples in $\partial\Gamma$ is a rather efficient source
of uniformity in our computations that seems likely to generalize well to more
complicated settings.

Acknowledgements. The author thanks Jean-Marc Schlenker for inspiring his
interest in questions related to this paper, Mike Wolf for his support and
interest, Aaron Calderon for stimulating conversations, and Christopher Bishop
for helpful comments on the exposition. This material is based on work
supported by the National Science Foundation under Grants No. 1842494 and
2005551.

##  2\. Simple Root Stretch Metrics

###  2.1. Teichmüller Space

We give a proof of Theorem A in the case of $\mathcal{T}(S)$ here, before
deducing the general case from [30]. This is a model of the core of the proof
of Theorem C.

####  2.1.1. Proximality

We briefly set notation and recall the standard results on proximal elements
of $\text{PSL}_{n}(\mathbb{R})$ that are essential to the main estimate. Our
notation follows [9].

A linear map $g\in\text{PSL}_{n}(\mathbb{R})$ is proximal if $g$ fixes a line
$x^{+}_{g}$ and transverse hyperplane $X^{<}_{g}$ so that
$\lim_{n\to\infty}g^{n}x=x^{+}_{g}$ for all $x\notin X^{<}_{g}$. With respect
to a background Riemannian metric $\delta$ on $\mathbb{R}\mathbb{P}^{n-1}$, we
say that for $r>\varepsilon>0$ the map $g$ is $(r,\varepsilon)$-proximal if

$\displaystyle\delta(x_{g}^{+},X^{<}_{g})\geq r,\qquad
g(\mathbb{RP}^{n-1}-N_{\varepsilon}(X^{<}_{g}))\subset
N_{\varepsilon}(x^{+}_{g}),\qquad
g|_{\mathbb{R}\mathbb{P}^{n-1}-N_{\varepsilon}(X^{<}_{g})}\text{ is
$\varepsilon$-Lipschitz.}$

where $N_{\varepsilon}(S)$ is the $\varepsilon$-neighborhood of a set $S$
throughout the paper. We say $g$ is $(r,\varepsilon)$-biproximal if both $g$
and $g^{-1}$ are $(r,\varepsilon)$-proximal. The consequence of [1] needed
here is:

######  Fact 2.1 ([1] 5.17, [3] §3.1).

Let $\rho_{a},\rho_{b}:\Gamma\to{\rm{PSL}}_{2}(\mathbb{R})$ be Fuchsian. Then
for any $r>\varepsilon>0$ sufficiently small there is a finite set
$F\subset\Gamma$ so that for all $\gamma\in\Gamma$, there is an $\eta\in F$ so
$\rho_{a}(\eta\gamma)$ and $\rho_{b}(\eta\gamma)$ are
$(r,\varepsilon)$-proximal.

####  2.1.2. Regularity Computation

The relevant notations to this computation are as follows. For Fuchsian
representations $\rho_{a},\rho_{b}$ and $\gamma\in\Gamma-\\{e\\}$, define
$r_{ab}(\gamma)=\ell_{\rho_{a}}(\gamma)/\ell_{\rho_{b}}(\gamma)$, and
$h_{ab}=\inf_{\gamma\in\Gamma-\\{e\\}}r_{ab}(\gamma)$. In [29], Thurston
proves
$d_{\text{Th}}(\rho_{a},\rho_{b})=\log\,\sup_{\gamma\in\Gamma-\\{e\\}}r_{ab}^{-1}$.
Denote the limit maps $\partial\Gamma\to\partial\mathbb{H}$ of $\rho_{a}$ and
$\rho_{b}$ by $\xi_{a}$ and $\xi_{b}$, respectively. For
$g\in\text{PSL}_{2}(\mathbb{R})$ let $\ell(g)$ be translation length on
$\mathbb{H}$.

######  Theorem 2.2 (Coupling is Stretch).

$d_{1}=d_{{\rm{Th}}}$ and $\phi_{ab}$ attains its optimal Hölder exponent.

###### Proof.

We first show $d_{1}\geq d_{\text{Th}}$. Let $\gamma\in\Gamma-\\{e\\}$ and
$\alpha>r_{ab}(\gamma)$. Work in models of $\mathbb{H}$ with attractor
$\xi_{c}(\gamma^{+})$ at $0$ and $\xi_{c}(\gamma^{-})$ at $\infty$ for
$c\in\\{a,b\\}$. The induced metric on $\mathbb{R}\mathbb{P}^{1}-\\{\infty\\}$
is multiplicatively comparable to the reference metric on
$\mathbb{R}\mathbb{P}^{1}$ in the complement of any fixed neighborhood of
$\infty$. Consider the sequence
$(x_{n},y_{n})=(e^{-n\ell_{\rho_{b}}(\gamma)},0)$. Then

$\displaystyle\frac{d(\phi_{ab}(x_{n}),\phi_{ab}(y_{n}))}{d(x_{n},y_{n})^{\alpha}}$
$\displaystyle=\frac{d(\rho_{a}(\gamma)^{n}\phi_{ab}(1),\rho_{a}(\gamma)^{n}\phi_{ab}(0))}{d(\rho_{b}(\gamma)^{n}1,\rho_{b}(\gamma)^{n}0)^{\alpha}}=e^{n(\alpha\ell_{\rho_{b}}(\gamma)-\ell_{\rho_{a}}(\gamma))}{d(\phi_{ab}(1),\phi_{ab}(0))}.$

As $\alpha>r_{ab}(\gamma)$, $\exp(n(\alpha\ell_{b}(\gamma)-\ell_{a}(\gamma))$
grows unboundedly. So $\phi_{ab}$ is not $\alpha$-Hölder. Hence

$d_{1}(\rho_{a},\rho_{b})=-\log\alpha_{ab}\geq-\log\left(\inf_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{a}}(\gamma)}{\ell_{\rho_{b}(\gamma)}}\right)=\log\left(\sup_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{b}}(\gamma)}{\ell_{\rho_{a}}(\gamma)}\right)=d_{\text{Th}}(\rho_{a},\rho_{b}).$

We prove the other inequality by showing the above behavior dominates the
critical cases for the Hölder condition. The key step is a use of Fact 2.1 to
separate attracting and repelling fixed-points of maps under consideration in
an application of the uniform convergence group property of $\Gamma$. The
other matter to control is that the metrics on
$\mathbb{R}\mathbb{P}^{1}-\\{\infty\\}$ induced from upper half plane models
of $\mathbb{H}$ have poor comparability near $\infty$.

Fix reference metrics $\delta$ on $\partial\mathbb{H}$ from the Poincaré model
and $d_{\partial\Gamma}$ on $\partial\Gamma$. We will exploit that only nearby
points are relevant to the Hölder condition since $\partial\Gamma$ is compact.
Let $r>\varepsilon>0$ be sufficiently small and fixed. By Fact 2.1, there is a
finite set $F\subset\Gamma$ so that for all $\gamma\in\Gamma$ there is an
$\eta\in F$ so that $\rho_{a}(\eta\gamma)$ and $\rho_{b}(\eta\gamma)$ are
$(r,\varepsilon)$-proximal.

As $\Gamma$ is a uniform convergence group, there is a compact subset $K$ of
the collection $\partial\Gamma^{(3)+}$ of positiviely oriented triples of
elements of $\partial\Gamma$ so that for all $t\in\partial\Gamma^{(3)+}$ there
is a $\gamma\in\Gamma$ so $\gamma t\in K$. Replacing $K$ with the compact set
$L=FK$, with

$\Gamma_{(r,\varepsilon)}^{a,b}=\\{\gamma\in\Gamma\mid\rho_{a}(\gamma)\text{
and }\rho_{b}(\gamma)\text{ are }(r,\varepsilon)\text{-proximal}\\},$

for every $t\in\partial\Gamma^{(3)+}$ there is a
$\gamma\in\Gamma_{(r,\varepsilon)}^{a,b}$ so $\gamma t\in L$. Compactness of
$L$ ensures that there is a $\nu>0$ so that for all $(x,y,z)\in L$ each of
$\delta(\xi_{a}(x),\xi_{a}(y)),\delta(\xi_{a}(x),\xi_{a}(z))$, and
$\delta(\xi_{a}(y),\xi_{a}(z))$ and their analogues for $\xi_{b}$ are at least
$\nu$.

We next use our freedom to choose the scale of points we consider to reduce
the $\varepsilon$ for which we must consider $(r,\varepsilon)$-proximal maps
while still working with the same compact $L$.

So let $x,y\in\partial\Gamma$ be distinct. We take $d(\xi_{a}(x),\xi_{a}(y))$
and $d(\xi_{b}(x),\xi_{b}(y))$ to be below a fixed small scale, that we shall
describe. The first requirement is that
$d_{\partial\Gamma}(x,y)<\text{diam}(\partial\Gamma)/3$.

After potentially interchanging $x$ and $y$, we may choose
$z\in\partial\Gamma$ so $(x,y,z)$ is positively oriented and both
$d_{\partial\Gamma}(x,z),d_{\partial\Gamma}(y,z)$ are at least
$\text{diam}(\partial\Gamma)/3$. Then both $\xi_{a}(z)$ and $\xi_{b}(z)$ are
separated by a $(x,y)$-independent constant $r_{0}$ from
$\xi_{a}(x),\xi_{a}(y)$ and $\xi_{b}(x),\xi_{b}(y)$, respectively. Now there
exists a point $(x_{0},y_{0},z_{0})\in L$ and
$\gamma_{xy}\in\Gamma_{(r,\varepsilon)}^{a,b}$ so that
$\gamma_{xy}(x,y,z)=(x_{0},y_{0},z_{0})$.

######  Claim 2.3 (Proximality Refinement).

For all $\varepsilon^{\prime}>0$ there is a $\tau(\varepsilon^{\prime})>0$ so
that for all distinct $x,y$ in $\partial\Gamma$ with
$d_{\partial\Gamma}(x,y)<\tau(\varepsilon^{\prime})$ both
$\rho_{a}(\gamma_{xy})$ and $\rho_{b}(\gamma_{xy})$ are
$(r,\varepsilon^{\prime})$-biproximal.

###### Proof.

Fix $r>0$. Straightforward computation in the Poincaré model of $\mathbb{H}$
shows the following two observations.111A detailed argument in greater
generality is given in §4.2.3. First, for all $\varepsilon^{\prime}>0$ there
is an $T<\infty$ so if $g\in\text{PSL}_{2}(\mathbb{R})$ is proximal, has
$\delta(x_{g}^{+},X_{g}^{<})\geq r$, and has $\ell(g)>T$ then $g$ is
$(r,\varepsilon^{\prime})$-proximal. In fact, symmetry of this condition under
inversion allows a conclusion of $(r,\varepsilon^{\prime})$-biproximality.
Second, for all $T>0$ there is a $B=B(T,r)$ so for every proximal
$g\in\text{PSL}_{2}(\mathbb{R})$ with $\delta(x_{g}^{+},X_{g}^{<})\geq r$ and
$\ell(g)\leq T$ the action of $g$ on $\partial\mathbb{H}$ is $B$-bilipschitz.

The optimal bilipschitz factor of $\rho_{c}(\gamma_{xy})$ $(c\in\\{a,b\\})$ is
bounded below by $\nu/\delta(\xi_{c}(x),\xi_{c}(y))$. So take
$\tau(\varepsilon^{\prime})$ so that if
$d_{\partial\Gamma}(x,y)<\tau(\varepsilon^{\prime})$ then both
$\delta(\xi_{a}(x),\xi_{a}(y))$ and $\delta(\xi_{b}(x),\xi_{b}(y))$ are less
than a constant $\eta(\varepsilon^{\prime})$ so small that the bilipshitz
factor lower bound of $\nu/\eta(\varepsilon^{\prime})$ for the action of
$\rho_{c}(\gamma_{xy})$ on $\partial\mathbb{H}$ implies
$(r,\varepsilon^{\prime})$-biproximality of $\rho_{c}(\gamma_{xy})$
($c\in\\{a,b\\}$). ∎

Now let $\varepsilon^{\prime}<\text{min}(r_{0}/6,\nu/6)$. We restrict our
attention further to $(x,y)$ with $\delta(\xi_{a}(x),\xi_{a}(y))$ and
$\delta(\xi_{b}(x),\xi_{b}(y))$ less than
$\text{min}(\tau(\varepsilon^{\prime}),\varepsilon^{\prime}/3)$, where
$\tau(\varepsilon^{\prime})$ is provided by the preceding claim.

So fix such a pair $(x,y)$ and fix $c\in\\{a,b\\}$. We record how the
$(r,\varepsilon^{\prime})$-proximality of $\rho_{c}(\gamma_{xy})$ constrains
the location of $\gamma_{xy}^{+}$ and $\gamma_{xy}^{-}$ relative to the other
points. We first note that since $\rho_{c}(\gamma_{xy})$ is
$(r,\varepsilon^{\prime})$-proximal, at least one of $\xi_{c}(x)$ and
$\xi_{c}(y)$ must be in $N_{\varepsilon^{\prime}}(\xi_{c}(\gamma_{xy}^{-}))$:
otherwise both $\xi_{c}(\gamma_{xy}x)$ and $\xi_{c}(\gamma_{xy}y)$ would be in
$N_{\varepsilon^{\prime}}(\xi_{c}(\gamma_{xy}^{+}))$ which is impossible
because $\varepsilon^{\prime}<\nu$. So both $\xi_{c}(x)$ and $\xi_{c}(y)$ are
contained in $N_{2\varepsilon^{\prime}}(y)$.

Observe also that since $\delta(\xi_{c}(x),\xi_{c}(z))>6\varepsilon^{\prime}$
that $\xi_{c}(z)\notin N_{\varepsilon^{\prime}}(\gamma_{xy}^{-})$. So
$\rho_{c}(\gamma_{xy}z)\in N_{\varepsilon^{\prime}}(\gamma_{xy}^{+})$ by
$(r,\varepsilon^{\prime})$-proximality of $\rho_{c}(\gamma_{xy})$. This
implies $\xi_{c}(\gamma_{xy}x)$ and $\xi_{c}(\gamma_{xy}y)$ are separated from
$\gamma_{xy}^{+}$ by at least $\nu/2$. We finally remark that for at least one
$w_{c}\in\\{x,y\\}$ the point $\xi_{c}(\gamma_{xy}w_{c})$ is separated from
$\xi_{c}(\gamma_{xy}^{-})$ by at least $\nu/2$, by construction of $\nu$.

Now, for each $c\in\\{a,b\\}$ work in an upper-half-space model of
$\mathbb{H}$, normalized so that $\xi_{c}(\gamma_{xy}^{-})$ is at $0$,
$\xi_{c}(\gamma_{xy}^{+})$ is at $\infty$, and $\xi_{c}(\gamma_{xy}w_{c})$ is
at $\pm 1$, with sign determined by the orientation of
$(\gamma_{xy}^{-},\gamma_{xy}w_{c},\gamma_{xy}^{+})$. From the uniform
separation of the normalization points, the metrics $d_{c}$ on
$\partial\mathbb{H}$ in these models are uniformly multiplicatively comparable
to $\delta$ on the complements of $\nu/2$-neighborhoods of
$\xi_{c}(\gamma_{xy}^{+})$ (taken with respect to $\delta$).

In these metrics,

$\displaystyle\frac{d_{a}(\phi_{ab}(\xi_{b}(x)),\phi_{ab}(\xi_{b}(y)))}{d_{b}(\xi_{b}(x),\xi_{b}(y))^{h_{ab}}}$
$\displaystyle=\frac{e^{-\ell_{a}(\gamma_{xy}^{-1})}d_{a}(\phi_{ab}(\xi_{b}(x_{0})),\phi_{ab}(\xi_{b}(y_{0})))}{e^{-h_{ab}\ell_{b}(\gamma_{xy}^{-1})}d_{b}(\xi_{b}(x_{0}),\xi_{b}(y_{0}))^{h_{ab}}}$
$\displaystyle\leq\exp{(h_{ab}\ell_{b}(\gamma_{xy}^{-1})-\ell_{a}(\gamma_{xy}^{-1}))}C$
$\displaystyle\leq C,$

for a fixed finite $C$ supplied by the compactness of $L$. So
$h_{ab}=\alpha_{ab}$ and $\phi_{ab}$ is $h_{ab}$-Hölder and

$\displaystyle
d_{1}(\rho_{a},\rho_{b})=-\log\alpha_{ab}\leq-\log\left(\inf_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{a}}(\gamma)}{\ell_{\rho_{b}(\gamma)}}\right)=\log\left(\sup_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{b}}(\gamma)}{\ell_{\rho_{a}}(\gamma)}\right)=d_{\text{Th}}(\rho_{b},\rho_{a}).$

###  2.2. Notations and reminders

We set some notation here for the rest of the paper.

####  2.2.1. Lie-Theoretic Notations

Let
$\mathfrak{a}=\\{(x_{1},...,x_{n})\in\mathbb{R}^{n}\mid\sum_{i=1}^{n}x_{i}=0\\}$
be a standard Cartan subalgebra of $\mathfrak{sl}_{n}(\mathbb{R})$, identified
with diagonal trace-free matricies in $\mathfrak{gl}_{n}(\mathbb{R})$. We fix
the Weyl chamber $\mathfrak{a}^{+}=\\{(x_{1},...,x_{n})\in\mathfrak{a}\mid
x_{1}\geq...\geq x_{n}\\}$. Denote the interior of $\mathfrak{a}^{+}$ by
$\mathring{\mathfrak{a}}^{+}$. Let
$\lambda:\text{PSL}_{n}(\mathbb{R})\to\mathfrak{a}^{+}$ be the Jordan
projection, i.e. $\lambda(g)=(\ell_{1}(g),...,\ell_{n}(g))$ is the logarithms
of the absolute values of the generalized complex eigenvalues of $g$, listed
in non-increasing order with multiplicity.

For $\varphi\in\mathfrak{a}^{*}$ so that the restriction of $\varphi$ to
$\mathfrak{a}^{+}$ is non-negative, the $\varphi$-length
$\ell^{\varphi}:\text{PSL}_{n}(\mathbb{R})\to\mathbb{R}^{+}$ is given by
$\ell^{\varphi}(g)=\varphi(\lambda(g))$. We adopt the notation that for a
representation $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ and
$\gamma\in\Gamma$ that
$\ell_{\rho}^{\varphi}(\gamma)=\ell^{\varphi}(\rho(\gamma))$.

For $i=1,...,n-1$ write the $i$-th simple root $\mathfrak{a}\to\mathbb{R}$
given by $(x_{1},...,x_{n})\mapsto x_{i}-x_{i+1}$ as $\alpha_{i}$. The Hilbert
functional $\text{H}:\mathfrak{a}\to\mathbb{R}$ is given by
$\text{H}(x_{1},...,x_{n})=x_{1}-x_{n}=\sum_{i=1}^{n-1}\alpha_{i}$.

The limit cone $\mathcal{L}_{\Lambda}$ of a subgroup
$\Lambda<\text{PSL}_{n}(\mathbb{R})$ is the smallest closed cone containing
$\lambda(g)$ for all $g\in\Lambda$ [3]. Denote the limit cone of the image
$\rho(\Gamma)$ of a representation $\rho$ by $\mathcal{L}_{\rho}$.

####  2.2.2. Hitchin Representations and Hyperconvexity

We recall the characterization of Hitchin representations in terms of the
structure of their boundary maps and some related facts.

Informally, the relevant condition on a map
$\xi:\partial\Gamma\to\mathcal{F}(\mathbb{R}^{n})$ is that subspaces formed as
sums of entries of $\xi(x)$ are in as general position as possible, and limits
of sums of flag entries for nearby points converge to higher entries of the
flag. Formally, a continuous map
$\xi:\partial\Gamma\to\mathcal{F}(\mathbb{R}^{n})$ is a hypercovnex Frenet
curve if:

  1. (1)

For any $k_{1},...,k_{j}$ so $\sum_{l=1}^{j}k_{l}=p\leq n$, and distinct
$x_{1},...,x_{j}\in\partial\Gamma$, the sum
$\xi^{k_{1}}(x_{1})+...+\xi^{k_{j}}(x_{j})$ is direct.

  2. (2)

For any $x\in\partial\Gamma$, $k_{1},...,k_{j}$ as above, and sequence
$(x_{1}^{m},...,x_{j}^{m})$ of $j$-ples of distinct points in $\partial\Gamma$
converging to $(x,...,x)$ we have that
$\xi^{p}(x)=\lim\limits_{m\to\infty}\bigoplus_{l=1}^{j}\xi^{k_{l}}(x_{l}^{m})$.

This condition implies far more families of subspaces than those in the
definition must be in general position and satisfy limit compatibility
conditions, as is shown in [10]. We use the results of [10] in the following,
especially Théorème 2, Lemme 6, and Prop. 7. We have:

######  Theorem 2.4 ([11] [15]).

A representation $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ is Hitchin if and
only if there exists a $\rho$-equivariant hyperconvex Frenét curve
$\xi_{\rho}:\partial\Gamma\to\mathcal{F}(\mathbb{R}^{n})$.

The structure of exterior powers of Hitchin representations is relevant in the
sequel. The definition here is due to Pozetti-Sambarino-Wienhard [24]: a
representation $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ is
(1,1,2)-hyperconvex if it is $\\{P_{1},P_{2}\\}$-Anosov (see [13]) and
$(\xi^{1}(x)\oplus\xi^{1}(y))\cap\xi^{n-2}(z)=\\{0\\}$ for all distinct
$x,y,z\in\partial\Gamma$. The Anosov condition gaurantees the existence of
limit maps $(\xi^{1},\xi^{2},\xi^{n-2},\xi^{n-1})$ of partial flags consisting
of subspaces of the dimensions labelled by superscripts. All that we shall
need on $(1,1,2)$-hyperconvexity is contained in [25]:

  1. (1)

If $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ is Hitchin, then
$\Lambda^{k}\rho$ is $(1,1,2)$-hyperconvex for $k=1,...,n-1$,

  2. (2)

If $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ is $(1,1,2)$-hyperconvex, then
$\xi^{1}(\partial\Gamma)\subset\mathbb{R}\mathbb{P}^{n-1}$ is $C^{1}$ with
tangent line $\xi^{2}(x)$ at $\xi^{1}(x)$ for all $x\in\partial\Gamma$.

###  2.3. General Simple Root Stretch Metrics

We explain how Theorem A follows from work of Tsouvalas. The observation made
here almost appears in §6 of [30], in particular around Corollary 6.1. His
theorem, stated in our notation, is:

######  Theorem 2.5 (Tsouvalas [30] Thm. 1.9).

Let $\rho_{a}$ and $\rho_{b}$ be $(1,1,2)$-hyperconvex representations to
${\rm{PSL}}_{n}(\mathbb{R})$. Then
$\psi_{ab}:\xi^{1}_{b}(\partial\Gamma)\to\xi^{1}_{a}(\partial\Gamma)$ defined
by $\xi^{1}_{a}\circ(\xi^{1}_{b})^{-1}$ realizes its supremal $\alpha$-Hölder
regularity, with Hölder exponent
$\alpha_{ab}=\inf_{\gamma\in\Gamma-\\{e\\}}\ell_{\rho_{a}}^{\alpha_{k}}(\gamma)/\ell_{\rho_{b}}^{\alpha_{k}}(\gamma).$

Write the first entry of the limit map of $\Lambda^{n}\rho$ as
$\xi^{1}_{\Lambda^{k}\rho}$. To describe $\xi^{1}_{\Lambda^{k}\rho}$, denote
the Plücker embedding
$\text{Gr}_{k}(\mathbb{R}^{n})\to\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})$ by
$\text{Pl}_{n,k}$. Then
$\xi^{1}_{\Lambda^{k}\rho}=\text{Pl}_{n,k}\circ\xi^{k}$ with
$\xi=(\xi^{1},...,\xi^{n-1})$ the limit map of $\rho$. The coupling
construction is:

######  Definition 2.6.

For $\rho_{a},\rho_{b}:\Gamma\to{\rm{PSL}}_{n}(\mathbb{R})$ Hitchin, define
$\psi_{ab}^{k}:\xi^{1}_{\Lambda^{k}\rho_{b}}(\partial\Gamma)\to\xi^{1}_{\Lambda^{k}\rho_{a}}(\partial\Gamma)$
by
$\xi^{1}_{\Lambda^{k}\rho_{a}}\circ(\xi^{1}_{\Lambda^{k}\rho_{b}})^{-1}={\rm{Pl}}_{n,k}\circ\phi_{ab}^{k}\circ{\rm{Pl}}_{n,k}^{-1}.$

The Hölder regularities of $\phi_{ab}^{k}$ and $\psi_{ab}^{k}$ agree because
Plücker embeddings are smooth embeddings. From the behavior of eigenvalues of
diagonal matricies under exterior powers and Tsouvalas’ theorem we conclude
$\alpha_{ab}^{k}=\inf_{\gamma\in\Gamma-\\{e\\}}\ell_{\rho_{a}}^{\alpha_{k}}(\gamma)/\ell_{\rho_{b}}^{\alpha_{k}}(\gamma)$
for Hitchin representations
$\rho_{a},\rho_{b}:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ and $k=1,...,n-1$.
Theorem A follows:

$\displaystyle
d_{k}(\rho_{a},\rho_{b})=-\log\alpha_{ab}^{k}=-\log\left(\inf_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{a}}^{\alpha_{k}}(\gamma)}{\ell_{\rho_{b}}^{\alpha_{k}}(\gamma)}\right)=\log\left(\sup_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{b}}^{\alpha_{k}}(\gamma)}{\ell_{\rho_{a}}^{\alpha_{k}}(\gamma)}\right)=d_{\text{Th}}^{\alpha_{k}}(\rho_{a},\rho_{b}).$

##  3\. Bouquets

This section develops the structure of the tori $T_{\rho}^{k}$ used for the
estimates in §4. The key points are that each $T_{\rho}^{k}$ has two canonical
foliations by Frenet curves, that
$T_{\rho}^{k}-\xi^{1}_{\rho}(\partial\Gamma)$ is $C^{1}$, and that
$T_{\rho}^{k}-\xi^{1}(\partial\Gamma)$ has an approximate-product structure.

###  3.1. First Features

Let $\rho:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ be a Hitchin representation
with $n>3$. We recall the notations that the limit map of $\rho$ is
$\xi=(\xi^{1},...,\xi^{n-1})$, that $\mathcal{K}_{n}=\\{2,...,\lfloor
n/2\rfloor\\}$, that $k_{n}=\lfloor n/2\rfloor$, where $\lfloor\cdot\rfloor$
is the floor function, that
$B_{n}=(\partial\Gamma^{2}_{1}\sqcup...\sqcup\partial\Gamma^{2}_{k_{n}})/\sim$
is the $n$-bouquet defined in the introduction, and
$\Phi_{\rho}^{k}:\partial\Gamma^{2}_{k}\to\mathbb{R}\mathbb{P}^{n-1}$ are the
individual-$\partial\Gamma^{2}$ maps. The maps $\Phi_{\rho}^{k}$ agree on all
equivalent points and so induce a map
$\Phi_{\rho}:B_{n}\to\mathbb{R}\mathbb{P}^{n-1}$, with image denoted by
$\mathcal{B}_{\rho}$. Denote the set of pairs of distinct elements in
$\partial\Gamma$ by $\partial\Gamma^{(2)}$ and the collection of triples of
positively oriented elements of $\partial\Gamma$ by $\partial\Gamma^{(3)+}$.

######  Lemma 3.1.

The map $\Phi_{\rho}:B_{n}\to\mathbb{R}\mathbb{P}^{n-1}$ is a homeomorphism
onto $\mathcal{B}_{\rho}$.

###### Proof.

Since $B_{n}$ is compact and Hausdorff, it suffices to show that $\Phi_{\rho}$
is a continuous injection. Continuity on each individual
$\partial\Gamma^{2}_{k}$ ($k=1,...,k_{n}$) is a direct consequence of the
limit compatibility property of hyperconvex Frenet curves from their
characterization in terms of intersections ([10] Lemme 6). This implies
$\Phi_{\rho}$ is continuous. Injectivity is from the general position property
of hyperconvex Frenet curves for intersections ([10] Lemme 6). ∎

The following lemma is quite useful in understanding the structure of
$\mathcal{B}_{\rho}$.

######  Lemma 3.2 (Frenet Restriction).

Let $\xi:\partial\Gamma\to\mathcal{F}(\mathbb{R}^{n})$ be a hyperconvex Frenet
curve. Fix $1<D<n$ and $t_{0}\in\partial\Gamma$. Then
$\xi_{t_{0},D}=(\xi^{1}_{t_{0},D},...,\xi^{D-1}_{t_{0},D}):\partial\Gamma\to\mathcal{F}(\xi^{D}(t_{0}))$
defined by

$\xi^{k}_{t_{0},D}(s)=\begin{cases}\xi^{n-D+k}(s)\cap\xi^{D}(t_{0})&s\neq
t_{0}\\\ \xi^{k}(t_{0})&s=t_{0}\end{cases}$

is a hyperconvex Frenet curve.

###### Proof.

This follows from the characterization of hyperconvex Frenet curves in terms
of intersections ([10] Lemme 6). Namely, if $k_{1},...,k_{j}$ are positive
integers with $\sum_{i=1}^{j}k_{i}=D$ and
$x_{1},...,x_{j}\in\partial\Gamma-\\{t_{0}\\}$ are distinct, since $\xi$ is a
hyperconvex Frenet curve

(3.1)
$\displaystyle\bigcap_{i=1}^{j}\xi_{t_{0},D}^{D-k_{i}}(x_{i})\subset\xi^{D}(t_{0})\cap\left(\bigcap_{i=1}^{j}\xi^{n-k_{i}}(x_{i})\right)=\\{0\\}.$

Note that this implies that the inclusion in Equation 3.1 is an equality. The
case where one $x_{j}\in\partial\Gamma$ is equal to $t_{0}$ is similar, so
that hyperconvexity is established.

Limit compatibility is a consequence of the limit compatibility of $\xi$ as
follows. Let $\sum_{i=1}^{j}k_{i}=k<D$ and let
$(x_{1}^{m},x_{2}^{m},...,x_{j}^{m})$ be a sequence of $j$-ples of distinct
entries of $\partial\Gamma$ converging to $(x,...,x)$ for some
$x\in\partial\Gamma$. If $x\neq t_{0}$, by the general position property for
$\xi_{t_{0},D}$ proved above, transversality of $\xi^{D}(t_{0})$ and
$\xi^{n-k}(x)$, and limit compatibility of $\xi$,

$\displaystyle\lim_{m\to\infty}\bigcap_{i=1}^{j}\xi^{D-k_{i}}_{t_{0},D}(x_{i}^{m})=\lim_{n\to\infty}\xi^{D}(t_{0})\cap\left(\bigcap_{i=1}^{j}\xi^{n-k_{i}}(x_{i}^{m})\right)=\xi^{D}(t_{0})\cap\xi^{n-k}(x)=\xi^{D-k}_{t_{0},D}(x).$

The case in which $t_{0}=x$ is similar, with limit compatibility of $\xi$
playing the role of transversality of $\xi^{D}(t_{0})$ and $\xi^{n-k}(x)$ in
the above. ∎

We remark that due to the Frenet Restriction Lemma, natural objects associated
to Hitchin representations in projective spaces have inductive aspects.
Structures from lower rank reappear in intersections with subspaces, but
without natural group actions. See §3.3.

###  3.2. Regularity

The tori $T_{\rho}^{k}$ have complicated behavior near the image of $\xi^{1}$,
but much better controlled behavior elsewhere. The first relevant definition
is:

######  Definition 3.3.

Let $\Xi^{1}_{\rho}\subset T_{\rho}^{k}$ denote $\xi^{1}(\partial\Gamma)$ and
$\Xi^{2}_{k}=T_{\rho}^{k}-\Xi^{1}_{\rho}$. Let
$\Xi^{2}_{\rho}\subset\mathcal{B}_{\rho}$ denote
$\mathcal{B}_{\rho}-\Xi^{1}_{\rho}$.

In this subsection, we prove a $C^{1}$-regularity proposition for
$\Xi^{2}_{\rho}$ and a local product structure lemma that is useful in §4. The
sources of our control over regularity are the hyperconvexity of $\xi$ and
that the curves $\xi^{k}$ ($k=1,...,n-1$) are $C^{1}$, considered as subsets
of $\text{Gr}_{k}(\mathbb{R}^{n})$.

######  Proposition 3.4 (Regular off Frenet).

For $\rho\in{\rm{Hit}}_{n}(S)$, any $p\in\Xi^{2}_{k}$ is a $C^{1}$ point of
$T_{\rho}^{k}$.

Writing $p=\xi^{k}(x)\cap\xi^{n-k+1}(y)$ for $x\neq y$ in $\partial\Gamma$,
the tangent space is given by

$\displaystyle{\rm{T}}_{p}T_{\rho}^{k}=(\xi^{k}(x)\cap\xi^{n-k+2}(y))+(\xi^{k+1}(x)\cap\xi^{n-k+1}(y)).$

We adopt the convention $\xi^{n}(x)=\mathbb{R}^{n}$ for the edge case $k=2$.

######  Remark.

The restriction to $\Xi^{2}_{k}$ is essential. In the case $k=2$ and $n=4$, a
corollary of [20] is that $\Xi^{2}_{k}$ is exactly the set of $C^{1}$ points
of $T_{\rho}^{2}$.

######  Remark.

Particular care is needed in the analysis and exact statements of regularity
results in the following, as many closely related natural maps to those we
consider fail to be $C^{1}$.

For instance, it is a consequence of Benoist’s Limit Cone Theorem [3] that
with respect to the $C^{1}$ structure on $\Xi^{2}_{k}$ from its embedding in
$\mathbb{R}\mathbb{P}^{n-1}$ and any product $C^{1}$ structure on
$\partial\Gamma^{(2)}$ the maps
$\Phi_{\rho}^{k}:\partial\Gamma^{(2)}\to\Xi^{2}_{k}$ are never $C^{1}$ for
non-Fuchsian Hitchin representations.

####  3.2.1. Differential of Intersection

In this paragraph, we recall the formula for the differential of the
intersection of two subspaces in general position.

For a finite-dimensional real vector space $V$ and $k,l<\text{dim}(V)$ with
$k+l>\text{dim}(V)$, let

$\mathscr{T}_{k,l}(V)=\\{(U,W)\in\text{Gr}_{k}(V)\times\text{Gr}_{l}(V)\mid
U+W=V\\}$

and let $\pitchfork:\mathscr{T}_{k,l}\to\text{Gr}_{l+k-n}(V)$ be the
intersection map: $\pitchfork(U,W)=U\cap W$. Recall the canonical model for
$\text{T}_{W}\text{Gr}_{k}(V)$ as $\text{Hom}(W,V/W)$. For
$(U,W)\in\mathscr{T}_{k,l}(V)$ and
$(\phi,\psi)\in\text{T}_{(U,W)}\mathscr{T}_{k,l}(V)$, we define
$\\{\phi+\psi\\}\in\text{Hom}(U\cap W,V/(U\cap W))=\text{T}_{U\cap
W}\text{Gr}_{k+l-n}(V)$ as follows.

Since $k+l>n$, any $v\in V$ may be written as $v=a(v)+b(v)$ with $a(v)\in U$
and $b(v)\in W$. Each of $a(v)$ and $b(v)$ are defined up to addition of
elements of $U\cap W$. Pick arbitrary lifts $\overline{\phi}:U\to V$ of $\phi$
and $\overline{\psi}:W\to V$ of $\psi$. Note
$b\left(\overline{\phi}(v)\right)$ and $a\left(\overline{\psi}(v)\right)$
($v\in V$) are well-defined up to addition of elements of $U\cap W$. So a map
$\\{\phi+\psi\\}:U\cap W\to V/(U\cap W)$ is specified by
$\\{\phi+\psi\\}(v)=\left[b\left(\overline{\phi}(v)\right)+a\left(\overline{\psi}(v)\right)\right]$
($v\in U\cap W$). Here, $[\cdot]$ is equivalence modulo $U\cap W$.

######  Lemma 3.5 (Intersection Derivative).

For all $(U,W)\in\mathscr{T}_{k,l}(V)$ and
$(\phi,\psi)\in{\rm{T}}_{(U,W)}\mathscr{T}_{k,l}(V)$,

(3.2) $\displaystyle D_{(U,W)}\pitchfork(\phi,\psi)=\\{\phi+\psi\\}.$

The proof is an exercise in differential geometry. One solution takes the
derivatives of projective duality and direct sum as maps on Grassmannians then
applies the chain rule.

####  3.2.2. Tangent Lines to Limit Curves

We now give an expression for the tangent line of the image of the limit map
$\xi^{k}(\partial\Gamma)\subset\text{Gr}_{k}(\mathbb{R}^{n})$ of a Hitchin
representation $\rho$. The proof is by a reduction to [25]. We use the group
invariance of the Frenet curve throughout the paragraph.

Let $\mathbb{P}\text{T}\text{Gr}_{k}(\mathbb{R}^{n})$ be the projective
tangent bundle of $\text{Gr}_{k}(\mathbb{R}^{n})$ with projection $\pi_{k}$ to
$\text{Gr}_{k}(\mathbb{R}^{n})$ and define

$\mathscr{L}_{\rho}^{k}=\\{(x,\phi)\in\partial\Gamma\times\mathbb{P}\text{T}\text{Gr}_{k}(\mathbb{R}^{n})\mid\pi_{k}(\phi)=\xi^{k}(x),\,\xi^{k-1}(x)=\text{ker}\,\phi,\,\text{Im}(\phi)\subset\xi^{k+1}(x)\\}.$

As the target $\xi^{k+1}(x)/\xi^{k}(x)$ of $\phi$ is one-dimensional, for each
$x\in\partial\Gamma$ there is a unique point of the form
$(x,\phi)\in\mathscr{L}_{\rho}^{k}$. Note $\mathscr{L}_{\rho}^{k}$ is closed
in $\partial\Gamma\times\mathbb{P}\text{T}\text{Gr}_{k}(\mathbb{R}^{n})$.

######  Proposition 3.6 (Tangent Lines).

For any $x\in\partial\Gamma$, the tangent line
$\mathbb{P}{\rm{T}}_{\xi^{k}(x)}\xi^{k}(\partial\Gamma)$ is the unique
$\phi_{x}\in\mathbb{P}{\rm{T}}_{\xi^{k}(x)}{\rm{Gr}}_{k}(\mathbb{R}^{n})$ so
$(x,\phi_{x})\in\mathscr{L}_{\rho}^{k}$.

###### Proof.

Since $\xi^{k}(\partial\Gamma)$ is an embedded $C^{1}$ submanifold of
$\text{Gr}_{k}(\mathbb{R}^{n})$ and $\mathscr{L}_{\rho}^{k}$ is closed, it
suffices to prove the claim for the dense set of $\partial\Gamma$ consisting
of attracting fixed points $\gamma^{+}$ of $\gamma\in\Gamma-\\{e\\}$.

So let $\gamma\in\Gamma-\\{e\\}$, and let $v_{1},...,v_{n}$ denote a basis of
eigenvectors of $\rho(\gamma)$, ordered by decreasing modulus of the
corresponding eigenvalues $\lambda_{1},\lambda_{2},...,\lambda_{n}$. Then
$\xi^{k}(\gamma^{+})$ is the span of $v_{1},...,v_{k}$. Let $\Lambda^{k}\rho$
denote the $k$-th exterior power of $\rho$, which is $(1,1,2)$ hyperconvex.
Denote the first two entries of the associated limit curve by
$\xi^{1}_{\Lambda^{k}},\xi^{2}_{\Lambda^{k}}$. By [25] Prop. 7.4,
$\xi^{1}_{\Lambda^{k}}(\partial\Gamma)$ is an embedded $C^{1}$ submanifold of
$\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})$ with
$\text{T}_{\xi^{1}_{\Lambda^{k}}(x)}\xi^{1}(\partial\Gamma)=\xi^{2}_{\Lambda^{k}}(x)$
for all $x\in\partial\Gamma$.

By the dynamics preservation of limit maps of Anosov representations,

(3.3) $\displaystyle\xi^{1}_{\Lambda^{k}}(\gamma^{+})$
$\displaystyle=\text{span}\\{v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k}\\},$
$\displaystyle\xi^{2}_{\Lambda^{k}}(\gamma^{+})$
$\displaystyle=\text{span}\\{v_{1}\wedge v_{2}\wedge\cdots\wedge
v_{k},v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k-1}\wedge v_{k+1}\\}.$

Since the Plücker embedding $\text{Pl}_{k,n}$ is a smooth embedding and
$\xi^{1}_{\Lambda^{k}}(x)=\text{Pl}_{n,k}(\xi^{k}(x))$ for
$x\in\partial\Gamma$, to establish the claim it suffices to show
$D_{\xi^{k}(\gamma^{+})}\text{Pl}_{n,k}(\phi_{\gamma^{+}})$ is contained in
$\xi^{2}_{\Lambda^{k}}(\gamma^{+})$.

A representative of the projective class of $\phi_{\gamma^{+}}$ is given on
the basis $v_{1},...,v_{n}$ of $\mathbb{R}^{n}$ by
$\phi_{\gamma^{+}}(v_{i})=\delta_{k,k+1}v_{j}$ in Kronecker $\delta$-notation.
A representative $\psi^{k}_{\gamma^{+}}$ of the equivalence class of
$\xi^{2}_{\Lambda^{k}}(\gamma^{+})\in\mathbb{P}\text{T}_{\xi^{1}_{\Lambda^{k}}(x)}\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})$
is given on the standard basis $\\{w_{I}\\}$ indexed by positively ordered
$k$-element subsets $I\subset(1,...,n)$ of $\Lambda^{k}\mathbb{R}^{n}$ induced
by the basis $v_{1},...,v_{n}$ of $\mathbb{R}^{n}$ by
$\psi^{k}_{\gamma^{+}}w_{I}=\delta_{I_{1}I_{2}}w_{J}$ where $I_{1}=(1,...,k)$
and $I_{2}=(1,...,{k-1},{k+1})$.

Now, $A_{t}=\text{span}(v_{1},...,v_{k-1},v_{k}+tv_{k+1})$ in
$\text{Gr}_{k}(\mathbb{R}^{n})$ is tangent to $\phi_{\gamma^{+}}$ at $t=0$.
For $t>0$,

$\displaystyle\text{Pl}_{n,k}(A_{t})=\text{span}\\{(v_{1}\wedge
v_{2}\wedge\cdots\wedge v_{k})+t(v_{1}\wedge v_{2}\wedge\cdots\wedge
v_{k-1}\wedge v_{k+1})\\}$

is tangent to $\xi^{2}_{\Lambda^{k}}(\gamma^{+})$, as desired. ∎

####  3.2.3. Ambient Regularity

We now address the regularity of $\Xi^{2}_{\rho}$.

###### Proof of Prop. 3.4.

Let $\mathscr{D}_{\rho}^{k}\subset\mathscr{T}_{k,n-k+1}$ be the submanifold of
$\xi^{k}(\partial\Gamma)\times\xi^{n-k+1}(\partial\Gamma)$ given by

$\displaystyle\mathscr{D}_{\rho}^{k}=\xi^{k}(\partial\Gamma)\times\xi^{n-k+1}(\partial\Gamma)-\\{(\xi^{k}(x),\xi^{n-k+1}(x))\mid
x\in\partial\Gamma\\},$

and let $\pitchfork_{\rho}^{k}$ be the restriction of $\pitchfork$ to
$\mathscr{D}_{\rho}^{k}$. Then $\Xi^{2}_{k}$ is the image of
$\pitchfork_{\rho}^{k}$. As $\pitchfork$ is smooth on $\mathscr{T}_{k,n-k+1}$
and $\mathscr{D}_{\rho}^{k}$ is $C^{1}$, to show that
$p=\xi^{k}(x)\cap\xi^{n-k+1}(y)$ $((x,y)\in\partial\Gamma^{(2)})$ is a $C^{1}$
point of $T_{\rho}^{k}$, it suffices to show that the differential
$D_{(\xi^{k}(x),\xi^{n-k+1}(y))}\pitchfork_{\rho}^{k}$ has rank $2$.

Given $(x,y)\in\partial\Gamma^{(2)}$ with corresponding points
$(x,\phi)\in\mathscr{L}_{\rho}^{k}$ and
$(y,\psi)\in\mathscr{L}_{\rho}^{n-k+1}$, the image of
$D_{(\xi^{k}(x),\xi^{n-k+1}(y))}\pitchfork_{\rho}^{k}$ is spanned by
$\\{\phi\\}$ and $\\{\psi\\}$. Note $\\{\phi\\}$ and $\\{\psi\\}$ each have
rank at most $1$ as $\phi$ and $\psi$ have rank $1$. By hyperconvexity of
$\xi$,

$\displaystyle\ker\phi\cap\xi^{n-k+1}(y)$
$\displaystyle=\xi^{k-1}(x)\cap\xi^{n-k+1}(y)=\\{0\\},$
$\displaystyle\ker\psi\cap\xi^{k}(x)$
$\displaystyle=\xi^{n-k}(y)\cap\xi^{k}(x)=\\{0\\}.$

We conclude that any lifts
$\overline{\phi},\overline{\psi}:p\to\mathbb{R}^{n}$ of $\phi|_{p}$ and
$\psi|_{p}$ have rank $1$. As $\text{Im}(\phi)\not\subset\xi^{k}(x)$ and
$\text{Im}(\psi)\not\subset\xi^{n-k+1}(y)$ the maps $\\{\phi\\}$ and
$\\{\psi\\}$ have nontrivial image and hence rank $1$.

By the definitions of $\mathscr{L}_{\rho}^{k}$, $\mathscr{L}_{\rho}^{n-k+1}$,
and $\\{\cdot\\}$, the image $\\{\phi\\}(p)$ is
$(\xi^{n-k+1}(y)\cap\xi^{k+1}(x))/p$ and the image $\\{\psi\\}(p)$ is
$(\xi^{k}(x)\cap\xi^{n-k+2}(y))/p$. As
$\mathbb{R}^{n}/p=(\xi^{k}(x)/p)\oplus(\xi^{n-k+1}(y)/p)$, the images of
$\\{\phi\\}$ and $\\{\psi\\}$ have nontrivial projections to distinct factors
of the direct sum decomposition of $\mathbb{R}^{n}/p$ and so do not coincide.
So $\\{\phi\\}$ and $\\{\psi\\}$ are linearly independent and
$D_{p}\pitchfork$ has rank $2$, as desired. The description of the tangent
space to $T_{\rho}^{k}$ at $p$ follows from our characterization of the images
of $\\{\phi\\}$ and $\\{\psi\\}$ in the above. ∎

####  3.2.4. Product Structure

The corollary of the regularity of $T_{\rho}^{k}$ that we use later is an
approximate-product metric model for compact subsets of $\Xi^{2}_{\rho}$. To
set notation, let $d_{\Xi^{2}}$ be the metric on $\Xi^{2}_{\rho}$ induced from
its embedding in $\mathbb{R}\mathbb{P}^{n-1}$. Let $d_{\text{Gr}}$ be the
product metric induced on
$\xi^{k}(\partial\Gamma)\times\xi^{n-k+1}(\partial\Gamma)$ by the factors’
embeddings in $\text{Gr}_{k}(\mathbb{R}^{n})$ and
$\text{Gr}_{n-k+1}(\mathbb{R}^{n})$.

######  Corollary 3.7 (Intersection Bilipschitz).

For any compact subset $K$ of $\Xi^{2}_{k}$, there is a constant $C_{K}>0$ so
that the map
$(\pitchfork_{\rho}^{k})^{-1}:K\to\xi^{k}(\partial\Gamma)\times\xi^{n-k+1}(\partial\Gamma)$
is $C_{K}$-bilipshitz onto its image with respect to the metrics $d_{\rm{Gr}}$
and $d_{\Xi}$.

###### Proof.

In the proof of Prop. 3.4, we saw $\pitchfork_{\rho}^{k}$ restricted to
$\mathscr{D}_{\rho}^{k}$ is an injective $C^{1}$ immersion. So the restriction
of $\pitchfork_{\rho}^{k}$ to any compact submanifold with boundary of
$\mathscr{D}_{\rho}^{k}$ is a $C^{1}$ embedding. So
$(\pitchfork_{\rho}^{k})^{-1}$ is bilipschitz on any compact set in
$\Xi^{2}_{k}$. ∎

###  3.3. Foliations and Sliding Maps

The tori $T_{\rho}^{k}$ have some further structure that we record here.
Namely, each torus $T_{\rho}^{k}$ has two canonical foliations by Frenet
curves and there are natural bijections between the leaves of these
foliations. We also establish some regularity features of the natural
bijections. The rest of the paper is independent of this subsection.

####  3.3.1. Definitions

The Frenet Restriction Lemma 3.2 results in the two natural foliations of
$\partial\Gamma^{2}$ mapping to foliations of $T_{\rho}^{k}$ by hyperconvex
Frenet curves, which we record:

######  Definition 3.8.

Let $\mathcal{G}_{k}$ be the foliation of $T_{\rho}^{k}$ by hyperconvex Frenet
curves in copies of $\mathbb{R}\mathbb{P}^{k-1}$ with leaves given for
$x\in\partial\Gamma$ by $g_{x}=\\{\Phi_{\rho}^{k}(x,y)\mid
y\in\partial\Gamma\\}$. Let $\mathcal{H}_{k}$ be the foliation of
$T_{\rho}^{k}$ by in copies of $\mathbb{R}\mathbb{P}^{n-k}$ with leaves given
for $x\in\partial\Gamma$ by $h_{x}=\\{\Phi_{\rho}^{k}(y,x)\mid
y\in\partial\Gamma\\}$.

There are distinguished bijections between leaves of $\mathcal{G}_{k}$ (resp.
$\mathcal{H}_{k}$), coming from the global product structure of
$T_{\rho}^{k}$. The case of $\mathcal{H}_{2}$ for $\text{PSL}_{4}(\mathbb{R})$
played a role in [21].

######  Definition 3.9 (Sliding Maps).

Let $x_{1},x_{2}\in\partial\Gamma$. Define ${\rm{sl}}^{\mathcal{G}}_{x_{1}\to
x_{2}}:g_{x_{1}}\to g_{x_{2}}$ with $g_{x_{1}},g_{x_{2}}$ leaves of
$\mathcal{G}_{k}$ by:

$\displaystyle{\rm{sl}}^{\mathcal{G}}_{x_{1}\to
x_{2}}:\xi^{k}(x_{1})\cap\xi^{n+1-k}(y)$
$\displaystyle\mapsto\xi^{k}(x_{2})\cap\xi^{n+1-k}(y)$ $\displaystyle y\neq
x_{1},x_{2}$ $\displaystyle\xi^{k}(x_{1})\cap\xi^{n+1-k}(x_{2})$
$\displaystyle\mapsto\xi^{1}(x_{2})$ $\displaystyle\xi^{1}(x_{1})$
$\displaystyle\mapsto\xi^{k}(x_{2})\cap\xi^{n+1-k}(x_{1}).$

Similarly, define ${\rm{sl}}^{\mathcal{H}}_{x_{1}\to x_{2}}:h_{x_{1}}\to
h_{x_{2}}$ with $h_{x_{1}}$, $h_{x_{2}}$ leaves of $\mathcal{H}_{k}$ by

$\displaystyle{\rm{sl}}^{\mathcal{H}}_{x_{1}\to
x_{2}}:\xi^{k}(y)\cap\xi^{n+1-k}(x_{1})$
$\displaystyle\mapsto\xi^{k}(y)\cap\xi^{n+1-k}(x_{2})$ $\displaystyle y\neq
x_{1},x_{2}$ $\displaystyle\xi^{k}(x_{2})\cap\xi^{n+1-k}(x_{1})$
$\displaystyle\mapsto\xi^{1}(x_{2})$ $\displaystyle\xi^{1}(x_{1})$
$\displaystyle\mapsto\xi^{k}(x_{1})\cap\xi^{n+1-k}(x_{2}).$

####  3.3.2. Sliding Regularity

We next address the regularity of sliding maps.

######  Lemma 3.10.

For $x_{1},x_{2}\in\partial\Gamma$ and $k\in\\{1,...,k_{n}\\}$ and
$\mathcal{K}\in\\{\mathcal{G}_{k},\mathcal{H}_{k}\\}$, the restriction of
${\rm{sl}}_{x_{1}\to x_{2}}^{\mathcal{K}}$ with domain $k_{x_{1}}$ to
$k_{x_{1}}-[(\Xi^{1}_{\rho}\cap k_{x_{1}})\cup({\rm{sl}}_{x_{1}\to
x_{2}}^{\mathcal{K}})^{-1}(\Xi^{1}_{\rho})]$ is $C^{1}$ with nonzero
derivative.

###### Proof.

We take $\mathcal{K}=\mathcal{G}_{k}$; the other case is analogous.

Since any hyperconvex Frenet curve
$\eta:\partial\Gamma\to\mathcal{F}(\mathbb{R}^{n})$ has $C^{1}$ image in
$\mathbb{R}\mathbb{P}^{n-1}$, the Frenet Restriction Lemma implies that both
$g_{x_{1}}$ and $g_{x_{2}}$ are embedded $C^{1}$ submanifolds of
$\mathbb{R}\mathbb{P}^{n-1}$.

For $x\in\partial\Gamma$, Let
$I_{x}^{k}:(\xi^{n-k+1}(\partial\Gamma)-\xi^{n-k+1}(x))\to g_{x}^{k}$ be
$\xi^{n-k+1}(y)\mapsto\xi^{n-k+1}(y)\cap\xi^{k}(x)$. The analysis of the proof
of Prop. 3.4 shows $I_{x}^{k}$ is $C^{1}$ with differential of full rank. As
$\text{sl}^{\mathcal{K}}_{x_{1}\to
x_{2}}=I_{x_{2}}^{k}\circ(I_{x_{1}}^{k})^{-1}$, the claim follows from the
chain rule. ∎

##  4\. Bouquet Coupling

We prove Theorems C and D in this section. Most of the discussion is on the
bounds in Theorem C, with the upper bound the main point. We prove the lower
bound in §4.1 and the upper bound in §4.2. We address completeness in §4.3. A
first observation is:

######  Lemma 4.1.

$D_{B}$ satisfies the triangle inequality, with possible value $\infty$.

###### Proof.

Optimal Hölder exponents are in $[0,1]$ and supermultiplicative under
composition. ∎

###  4.1. Lower Bounds

We now prove the lower bounds for $D_{B}$ in Theorem C by examining explict
sequences of points as in §2.1 (c.f. also [4]).

The following is the general form of the upper bound on Hölder exponents that
we prove and use as in §2.1. It is relevant to us because of the Frenet
Restriction Lemma.

######  Lemma 4.2.

Suppose $V$ is a real vector space of dimension $m$ and $g,h\in{\rm{PGL}}(V)$
are real-diagonalizable with $\lambda(g)=(\ell_{1}(g),...,\ell_{m}(g))$ and
$\lambda(h)=(\ell_{1}(h),...,\ell_{m}(h))$ each in
$\mathring{\mathfrak{a}}^{+}$. Let $v_{1}(g),...,v_{m}(g)$ and
$v_{1}(h),...,v_{m}(h)$ be eigenvectors corresponding to
$\ell_{1}(g),...,{\ell_{m}(g)}$ and $\ell_{1}(h),...,\ell_{m}(h)$,
respectively.

Suppose $M_{1}$ and $M_{2}$ are $C^{1}$ circles in $\mathbb{P}(V)$ so
$[v_{1}(g)]\in M_{1}$ with
${\rm{T}}_{[v_{1}(g)]}M_{1}=[v_{1}(g)]\oplus[v_{2}(g)]$ and $[v_{1}(h)]\in
M_{2}$ with ${\rm{T}}_{[v_{1}(h)]}M_{2}=[v_{1}(h)]\oplus[v_{2}(h)]$. Let
$\phi:M_{1}\to M_{2}$ be a homeomorphism so that $\phi([v_{1}(g)])=[v_{1}(h)]$
and $\phi(gp)=h\phi(p)$ for all $p\in M_{1}$.

Then $\phi$ is not $\alpha$-Hölder for any
$\alpha>\ell^{\alpha_{1}}(h)/\ell^{\alpha_{1}}(g)$.

###### Proof.

Take affine charts $\mathcal{A}_{g}$ and $\mathcal{A}_{h}$ for $\mathbb{P}(V)$
so that in $\mathcal{A}_{g}$:

  1. (1)

$[v_{1}(g)]$ is at the origin,

  2. (2)

$[\text{span}(v_{2},...,v_{m})]$ is the hyperplane at infinity,

  3. (3)

For $j=2,...,m$, the $x_{j-1}$ axis is the intersection of
$[\text{span}(v_{1},v_{j})]$ with $\mathcal{A}_{g}$.

Arrange for the same conditions for $h$ to hold on $\mathcal{A}_{h}$. Outside
of a neighborhood of the hyperplane at infinity, the metrics in
$\mathcal{A}_{g}$ and $\mathcal{A}_{h}$ are multiplicatively comparable to a
reference metric on $\mathbb{P}(V)$. Let $x_{1}^{g}$ and $x_{1}^{h}$ be the
coordinate functions of $\mathcal{A}_{g}$ and $\mathcal{A}_{h}$.

Consider the following sequence of points $(z_{n},w_{n})\in M_{1}$. Take
$z_{n}=[v_{1}(g)]$ for all $n$. From the hypothesis on the tangent line to
$M_{1}$ at $[v_{1}(g)]$ take $w_{0}$ so that there is an interval
$I_{g}\subset M_{1}$ with endpoints $[v_{1}(g)]$ and $w_{0}$ so that
$\frac{1}{2}|x_{1}^{g}(w)|\leq d_{\mathcal{A}_{g}}(0,w)\leq 2|x_{1}^{g}(w)|$
for all $w\in I_{g}$, and the analogous claim holds for $h$ in
$\mathcal{A}_{h}$. Take $w_{n}=g^{n}w_{0}$ for all $n>0$.

Now let $\alpha>\ell^{\alpha_{1}}(h)/\ell^{\alpha_{1}}(g)$. Then for an
$n$-independent constant $C$,

$\displaystyle\frac{d_{\mathcal{A}_{h}}(\phi(z_{n}),\phi(w_{n}))}{d_{\mathcal{A}_{g}}(z_{n},w_{n})^{\alpha}}$
$\displaystyle=\frac{d_{\mathcal{A}_{h}}(0,h^{n}\phi(w_{0}))}{d_{\mathcal{A}_{g}}(0,g^{n}w_{0})^{\alpha}}\geq\frac{|x_{1}^{h}(h^{n}\phi(w_{0}))|}{4|x_{1}^{g}(g^{n}w_{0})|^{\alpha}}=\frac{e^{-n\ell^{\alpha_{1}}(h)}|x_{1}^{h}(\phi(w_{0}))|}{4e^{-\alpha
n\ell^{\alpha_{1}}(g)}|x_{1}^{g}(w_{0})|^{\alpha}}=\frac{e^{n\alpha\ell^{\alpha_{1}}(g)}}{e^{\ell^{\alpha_{1}}(h)}}C,$

which grows arbitrarily large. So $\phi$ is not $\alpha$-Hölder. ∎

We now establish our lower bounds for $D_{B}$. Let us set notation. For
Hitchin representations
$\rho_{a},\rho_{b}:\Gamma\to\text{PSL}_{n}(\mathbb{R})$ and
$\gamma\in\Gamma-\\{e\\}$, define
$r_{ab}^{k}(\gamma)=\ell^{\alpha_{k}}_{\rho_{a}}(\gamma)/\ell^{\alpha_{k}}_{\rho_{b}}(\gamma)$,
and let $h_{ab}^{k}=\inf_{\gamma\in\Gamma-\\{e\\}}r_{ab}^{k}(\gamma)$. A
useful remark is that $h_{ab}^{k}=h_{ab}^{n-1-k}$ because
$\ell_{\rho}^{\alpha_{k}}(\gamma)=\ell_{\rho}^{\alpha_{n-1-k}}(\gamma^{-1})$.

######  Proposition 4.3.

For $n>3$ and $\rho_{a},\rho_{b}\in{\rm{Hit}}_{n}(S)$,
${\rm{max}}_{i=1,...,n-1}d_{\rm{Th}}^{\alpha_{k}}(\rho_{a},\rho_{b})\leq
D_{B}(\rho_{a},\rho_{b})$.

###### Proof.

Denote by $\alpha_{ab}$ be the supremum of the $\alpha>0$ so that $\Phi_{ab}$
is $\alpha$-Hölder.

Let $\gamma\in\Gamma-\\{e\\}$. Lemma 4.2 applied to
$\xi^{1}_{\rho_{b}}(\partial\Gamma)$ and $\xi^{1}_{\rho_{a}}(\partial\Gamma)$
at the points $\xi^{1}_{\rho_{b}}(\gamma^{+})$ and
$\xi^{1}_{\rho_{a}}(\gamma^{+})$ with the map $\Phi_{ab}$ shows that
$\alpha_{ab}\leq r_{ab}^{1}(\gamma)$, so that $\alpha_{ab}\leq
h_{ab}^{1}(\gamma)$.

Next fix $k\in 2,...,k_{n}$. Consider the curves $\eta_{\gamma^{-},n-k+1}$,
and $\nu_{\gamma^{-},n-k+1}$ induced from the Frenet Restriction Lemma by
$\xi_{\rho_{b}}$ and $\xi_{\rho_{a}}$, respectively. Let
$M_{b}=\eta_{\gamma^{-},n-k+1}(\partial\Gamma)$ and
$M_{a}=\nu_{\gamma^{-},n-k+1}(\partial\Gamma)$. Then
$\rho_{c}(\gamma)|_{\xi^{n-k+1}(\gamma^{-})}$ has eigenvalues
$v_{k}(\rho_{c}(\gamma)),v_{k+1}(\rho_{c}(\gamma)),...,v_{n}(\rho_{c}(\gamma))$
for $c\in\\{a,b\\}$. Additionally,
$\eta^{1}_{\gamma^{-},n-k+1}(x)=\Phi_{\rho_{b}}^{k}(x,\gamma^{-})$ and
$\nu^{1}_{\gamma^{-},n-k+1}(x)=\Phi_{\rho_{a}}^{k}(x,\gamma^{-})$ for
$x\in\partial\Gamma$. We focus on the points
$\eta^{1}_{\gamma^{-},n-k+1}(\gamma^{+})=[v_{k}(\rho_{b}(\gamma))]$ and
$\nu^{1}_{\gamma^{-},n-k+1}(\gamma^{+})=[v_{k}(\rho_{a}(\gamma))]$.

Since any hyperconvex Frenet curve $\xi=(\xi^{1},\xi^{2},...,\xi^{l})$ has
$C^{1}$ image of $\xi^{1}$ in projective space with tangent line $\xi^{2}(x)$
at $\xi^{1}(x)$ for all $x\in\partial\Gamma$, the tangent lines to $M_{c}$ at
$[v_{k}(\rho_{c}(\gamma))]$ are
$[v_{k}(\rho_{c}(\gamma))]\oplus[v_{k+1}(\rho_{c}(\gamma))]$ for
$c\in\\{a,b\\}$. Lemma 4.2 applies to the restriction of $\Phi_{ab}$ to these
curves, and produces an upper bound $\alpha_{ab}\leq r_{ab}^{k}$, so that
$\alpha_{ab}\leq h_{ab}^{k}$.

So $\max_{k=1,...,k_{n}}d_{\text{Th}}^{\alpha_{k}}(\rho_{a},\rho_{b})\leq
D_{B}(\rho_{a},\rho_{b})$, and the equality $h_{ab}^{k}=h_{ab}^{n-1-k}$ gives
the claim. ∎

###  4.2. Upper Bounds

What remains to be shown in Theorem C is the upper bound on $D_{B}$. The
strategy roughly parallels the computation of lower bounds for Hölder
constants in §2.1, with adaptations to the technical challenges of our
setting. In particular, we split our analysis for points close to and far away
from the image of $\xi^{1}$. See §1.4.

####  4.2.1. Estimates Away from the Frenet Curve

Away from $\Xi^{1}_{\rho}$, the regularity of $\Xi^{2}_{\rho}$ allows for a
reduction to the regularity of the curves $\xi^{k}(\partial\Gamma)$ in
Grassmannians. Put

(4.1)
$\displaystyle\alpha_{ab}^{L}=\min_{k=1,...,n-1}\left(\inf_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{a}}^{\alpha_{k}}(\gamma)}{\ell_{\rho_{b}}^{\alpha_{k}}(\gamma)}\right)$

######  Lemma 4.4.

Let $\rho_{a},\rho_{b}\in{\rm{Hit}}_{n}(S)$ and $K$ be a compact subset of
$\Xi^{2}_{\rho_{b}}$. Then the restriction of $\Phi_{ab}$ to $K$ is
$\alpha_{ab}^{L}$-Hölder.

###### Proof.

From Lemma 3.1 it follows that $K$ is a disjoint union $K_{1}\sqcup...\sqcup
K_{k_{n}}$ of compact subsets of $\Xi_{k}^{2}$. It suffices to prove the
result for the individual compact sets $K_{k}$.

So let $k\in\\{2,...,k_{n}\\}$ be given. Denote by
$\phi_{ab}^{k}\times\phi_{ab}^{n-k+1}$ the product map
$\xi^{k}_{\rho_{b}}(\partial\Gamma)\times\xi^{n-k+1}_{\rho_{b}}(\partial\Gamma)\to\xi^{k}_{\rho_{a}}(\partial\Gamma)\times\xi^{n-k+1}_{\rho_{a}}(\partial\Gamma)$
induced by $\phi_{ab}^{k}$ and $\phi_{ab}^{n-k+1}$. Note
$\phi_{ab}^{k}\times\phi_{ab}^{n-k+1}$ is $\alpha_{ab}^{L}$-Hölder because the
optimal Hölder exponent of a product map is the minimum of the optimal Hölder
exponents of its factors. Also, by definition,
$\Phi_{ab}=\pitchfork_{\rho_{a}}^{k}\circ(\phi_{ab}^{k}\times\phi_{ab}^{n-k+1})\circ\pitchfork_{\rho_{b}}^{-1}$
on $K_{k}$. Since Hölder regularity is invariant under bilipschitz mappings,
Cor. 3.7 here and Theorem 1.9 of [30] show that $\Phi_{ab}$ restricted to
$K_{k}$ is $\alpha_{ab}^{L}$-Hölder. ∎

######  Remark.

Giving a direct proof of Lemma 4.4 with the standard approach in the
literature for similar estimates (e.g. [9], [30], the below) seems quite
challenging because of the presence of fixed-points of $\rho_{a}(\gamma)$ and
$\rho_{b}(\gamma)$ $(\gamma\in\Gamma)$ of intermediate-magnitude eigenvalues
in $\Xi^{2}_{\rho}$. In particular, the regularity of $\Xi^{2}_{\rho}$ appears
essential to the tractability of the above estimate.

####  4.2.2. Loxodromy

In this paragraph, we recall the relevant improvement to proximality to our
setting and some useful lemmas from work of Guichard [9].

Loxodromy is an improvement to proximality that includes all eigenvalue gaps.
We say that $g\in\text{GL}_{n}(\mathbb{R})$ is loxodromic if $g$ is
diagonalizable over $\mathbb{R}$ with eigenvalues
$\lambda_{1},...,\lambda_{n}$ of $g$ listed with multiplicity, and
$|\lambda_{i}|,|\lambda_{j}|$ are distinct for all $i\neq j$. Hitchin
representations $\rho\in\text{Hit}_{n}(S)$ have the property that
$\rho(\gamma)$ is loxodromic for all $\gamma\in\Gamma-\\{e\\}$ [14].

We adopt the convention that $\lambda_{1},...,\lambda_{n}$ are ordered by
decreasing modulus, and the corresponding vectors are labelled
$v_{1},...,v_{n}$ and are taken to have norm $1$ with respect to a background
inner product on $\mathbb{R}^{n}$. We also write $\ell_{i}=\log|\lambda_{i}|$
for $i=1,...,n$. If $g$ is loxodromic, then for all $k=1,...,n-1$ the $k$-th
exterior power $\Lambda^{k}g$ acts proximally on $\mathbb{P}(\Lambda^{k}V)$.
For $g$ loxodromic, the map $\lambda:g\mapsto(\ell_{1}(g),...,\ell_{n}(g))$
coincides with the Jordan projection $\lambda$.

Let $\delta_{0}$ be a metric induced on $\mathbb{P}(V)$ by an inner product on
$V$, from realizing $\mathbb{P}(V)$ as the quotient of the unit sphere in $V$
by negation. Then the inner product inducing $\delta_{0}$ induces an inner
product on $\Lambda^{k}V$ and hence metrics $\delta_{0}^{k}$ on
$\mathbb{P}(\Lambda^{k}V)$ for $k=1,...,n-1$. If $g$ is loxodromic, with
eigenspaces $[v_{1}],[v_{2}],...,[v_{n}]$, taking norm-$1$ representatives
$v_{i}$ with respect to the background inner product, then specifying a new
inner product in which $\\{v_{i}\\}_{1,...,n}$ is orthonormal produces new
metrics $\delta_{g}$ on $\mathbb{P}(V)$ and $\delta_{g}^{k}$ on
$\mathbb{P}(\Lambda^{k}V)$ for all $k=1,...,n-1$.

We say $g\in\text{GL}(V)$ is $(r,\varepsilon)$-loxodromic if $\Lambda^{k}g$ is
$(r,\varepsilon)$-proximal with respect to $\delta_{0}^{k}$ for all
$k=1,...,n-1$. If, furthermore, $g^{-1}$ is $(r,\varepsilon)$-loxodromic, we
say that $g$ is $(r,\varepsilon)$-biloxodromic. The following is a standard
consequence of Abels-Margulis-Soifer’s work [1]:

######  Fact 4.5 (Corollary of [1]).

Let $\rho_{a},\rho_{b}$ be Hitchin, let $r>0$ be sufficiently small, and let
$r>\varepsilon>0$. Then there is a finite set $F\subset\Gamma$ so that for all
$\gamma\in\Gamma$ there is some $\eta\in F$ so that $\rho_{a}(\eta\gamma)$ and
$\rho_{b}(\eta\gamma)$ are $(r,\varepsilon)$-loxodromic.

####  4.2.3. Loxodromy Refinement

Our first task in analyzing $\Phi_{ab}$ near $\Xi^{1}$ is to prove the
following general analogue of the Proximality Refinement Claim 2.3. We adopt
the notation that $B_{n}^{(2)}=\\{(p,q)\in B_{n}^{2}\mid p\neq q\\}$.

######  Proposition 4.6 (Loxodromy Refinement).

Let $\rho:\Gamma\to{\rm{PSL}}_{n}(\mathbb{R})$ ($n\geq 2$) be Hitchin, $r>0$
be sufficiently small, and $r>\varepsilon>0$. Let
$\Gamma^{\rho}_{(r,\varepsilon)}$ be the collection of $\gamma\in\Gamma$ so
that $\rho(\gamma)$ is $(r,\varepsilon)$-loxodromic, and let $K\subset
B_{n}^{(2)}$ be compact.

Then for any $\varepsilon^{\prime}>0$ there is a
$\tau(\varepsilon^{\prime})>0$ so that for all
$\gamma\in\Gamma^{\rho}_{(r,\varepsilon)}$ if there exists $(p,q)\in
B_{n}^{(2)}$ with both
$\delta_{0}(\Phi_{\rho}(p),\Phi_{\rho}(q))<\tau(\varepsilon^{\prime})$ and
$(\gamma p,\gamma q)\in K$ then $\rho(\gamma)$ is
$(r,\varepsilon^{\prime})$-biloxodromic.

We begin with preliminaries. We shall repeatedly use the following:

######  Proposition 4.7 (Guichard [9] Prop. 18).

Let $r>0$ be sufficiently small. There is a compact subset
$K_{r}\subset{\rm{GL}}_{n}(\mathbb{R})$ so that for any $\varepsilon\in(0,r)$
and $(r,\varepsilon)$-loxodromic $g\in{\rm{GL}}_{n}(\mathbb{R})$ with norm-$1$
representatives $v_{1},...,v_{n}$ of the eigenslines of $g$, and with
$e_{1},...,e_{n}$ a fixed orthonormal basis of the background inner product,
there is a $T(g)\in K_{r}$ so that
$(v_{1},...,v_{n})^{T}=T(g)(e_{1},...,e_{n})^{T}$.

The below is a corollary of the preceding ([9] Cor. 19) and the observation
that inducing metrics on exterior products is natural with respect to change
of basis.

######  Corollary 4.8 (Guichard [9]).

For any $r>0$ there is a $C_{r}>0$ so for all $\varepsilon\in(0,r)$,
$k\in\\{1,...,n-1\\}$, and any $(r,\varepsilon)$-proximal $g$, the metrics
$\delta_{g}^{k}$ and $\delta_{0}^{k}$ are $C_{r}$-multiplicatively comparable.
That is, $C_{r}^{-1}\delta_{g}^{k}(x,y)\leq\delta_{0}^{k}(x,y)\leq
C_{r}\delta_{g}^{k}(x,y)$ for all
$x,y\in\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})$.

Prop. 4.6 uses the following link between eigenvalue gaps and quantitative
proximality.

######  Lemma 4.9.

Fix a closed cone $L$ contained in $\mathring{\mathfrak{a}}^{+}$ and
$r>\varepsilon>0$ sufficiently small. For any $\varepsilon^{\prime}>0$ there
is a $T>0$ so that if $g\in{\rm{PSL}}_{n}(\mathbb{R})$ is
$(r,\varepsilon)$-loxodromic and satisfies:

  1. (1)

(Jordan Projection Control) $\lambda(g)\in L$,

  2. (2)

(Hilbert Length Control) ${\rm{H}}(g)>T$,

then $g$ is $(r,\varepsilon^{\prime})$-biloxodromic.

###### Proof.

We may replace $L$ with $L\cup\iota(L)$, with $\iota$ the opposition
involution. As $\lambda(g^{-1})=\iota(\lambda(g))$ and
$H(\lambda(g^{-1}))=H(\lambda(g))$, it suffices to prove
$(r,\varepsilon^{\prime})$-loxodromy instead of biloxodromy.

Let us begin by noting that since $L$ is a closed cone in
$\mathring{\mathfrak{a}}^{+}$ there is a constant $c_{L}>0$ so
$\alpha_{k}(x)>c_{L}\text{H}(x)$ for all $x\in L$ and $k\in\\{1,...,n-1\\}$.
So it suffices to take $\ell^{\alpha_{k}}(g)>T$ for all $k\in\\{1,...,n-1\\}$
in our analysis. As the desired property is invariant under replacing the
metrics $\delta_{0}^{k}$ on $\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})$ with
uniformly multiplicatively comparable metrics, we may replace $\delta_{0}^{k}$
on $\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})$ by $\delta_{g}^{k}$ for all
$k\in\\{1,...,n-1\\}$.

We now compute. Let $k\in\\{1,...,n-1\\}$ be fixed. Let
$[v_{1}^{k}],...,[v_{N}^{k}]$ ($N=\binom{n}{k}$) be the eigenlines of
$\Lambda^{k}g$, ordered with nondecreasing modulus of eigenvalues, and with
corresponding eigenvalues $\lambda_{1}^{k},...,\lambda_{N}^{k}$. The largest-
modulus eigenvalue eigenspace $[v_{1}^{k}]$ of $\Lambda^{k}g$ is
$[v_{1}\wedge...\wedge v_{k}]$ with corresponding eigenvalue
$\lambda^{k}_{1}=\lambda_{1}...\lambda_{k}$. Note
$|\lambda^{k}_{1}/\lambda^{k}_{j}|\geq|\lambda_{k}/\lambda_{k+1}|>1$ for all
$j>1$.

Let $\mathcal{A}$ be an affine chart with $X_{\Lambda^{k}g}^{<}$ the plane at
infinity, $x_{\Lambda^{k}g}^{+}$ the origin, and the $x_{j}$-axis
($j=1,...,N-1$) given by the intersection of the projective line
$x_{g}^{+}\oplus\mathbb{R}{v_{j+1}^{k}}$ for $1\leq j\leq N-1$, and with a
point that is uniformly separated from all hyperplanes given by the span of
$N-1$ eigenlines of $\Lambda^{k}g$ placed at $(1,...,1)$.

Fix $\varepsilon^{\prime}>0$. Then there are
$C_{\varepsilon^{\prime}},R_{\varepsilon^{\prime}},r_{\varepsilon^{\prime}}>0$
so the Euclidean metric $d_{\mathcal{A}}$ on $\mathcal{A}$ satisfies:

  1. (1)

The restriction of $d_{\mathcal{A}}$ to
$\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})-N_{\varepsilon^{\prime}}(X_{\Lambda^{k}g}^{<})$
has $C_{\varepsilon^{\prime}}^{-1}\delta_{g}^{k}\leq d_{\mathcal{A}}\leq
C_{\varepsilon^{\prime}}\delta_{g}^{k}$,

  2. (2)

$\mathbb{P}(\Lambda^{k}\mathbb{R}^{n})-N_{\varepsilon^{\prime}}(X_{\Lambda^{k}g}^{<})$
is contained in $N_{R_{\varepsilon}^{\prime}}(0)$ in $\mathcal{A}$ with
respect to $d_{\mathcal{A}}$,

  3. (3)

$N_{\varepsilon^{\prime}}(x_{\Lambda^{k}g}^{+})$ contains
$N_{r_{\varepsilon^{\prime}}}(0)$ in $\mathcal{A}$ with respect to
$d_{\mathcal{A}}$.

Now, $\Lambda^{k}g$ acts on $\mathcal{A}$ by

$g[x_{1}:x_{2}:...:x_{n-1}:1]=\left[(\lambda_{2}^{k}/\lambda_{1}^{k})x_{1}:(\lambda_{3}^{k}/\lambda_{1}^{k})x_{2}:...:\left(\lambda_{N}^{k}/\lambda_{1}^{k}\right)x_{{N}-1}:1\right].$

This map is Lipschitz with Lipschitz constant
$\lambda_{2}^{k}/\lambda_{1}^{k}=\exp(-\alpha_{k}(g))$, and maps $N_{R}(0)$ to
$N_{\lambda_{2}^{k}R/\lambda_{1}^{k}}(0)$ for all $R>0$. The claim follows. ∎

We now prove Prop. 4.6. The point is that in the model metric $\delta_{g}$ the
differential of $g$ can be controlled in terms of the Hilbert length.

###### Proof of Prop. 4.6.

Let $n,K,\rho,r,\varepsilon,$ and $\varepsilon^{\prime}$ be given as stated.
Since limit cones of Hitchin representations are contained
$\mathring{\mathfrak{a}}^{+}$ ([23] Prop. 4.4), we may apply Lemma 4.9 with
$\mathcal{L}_{\rho(\gamma)}$. In particular, Lemma 4.9 shows there is a
$T<\infty$ so that it suffices to produce a $\tau$ so that if
$\delta_{0}(\Phi_{\rho}(p),\Phi_{\rho}(q))<\tau$ and $(\gamma p,\gamma q)\in
K$ with $\gamma\in\Gamma_{(r,\varepsilon)}^{\rho}$ then
$\ell^{\text{H}}_{\rho}(\gamma)>T$. By Corollary 4.8, the identity maps
between the metrics $\delta_{\rho(\gamma)}$ and $\delta_{0}$ for all $\gamma$
under consideration are uniformly bilipschitz. So we are free to work with the
metrics $\delta_{\rho(\gamma)}$ on $\mathbb{R}\mathbb{P}^{n-1}$ throughout the
following.

######  Claim 4.10.

Let $r>0$ be sufficiently small. For every $T>0$ there is a
$B=B(T,r,\rho)<\infty$ so if $\gamma\in\Gamma^{\rho}_{r,\varepsilon}$ for any
$\varepsilon<r$ and $\ell^{\rm{H}}_{\rho}(\gamma)\leq T$ then the action of
$\rho(\gamma)$ on $(\mathbb{R}\mathbb{P}^{n-1},\delta_{\rho(\gamma)})$ is
$B$-bilipschitz.

###### Proof.

Since we work with $\delta_{\rho(\gamma)}$, we may treat $\rho(\gamma)$ as
diagonal with respect to a fixed basis of $\mathbb{R}^{n}$. The collection
$\mathscr{C}_{\rho,T}$ of elements $h$ of $\text{PGL}_{n}(\mathbb{R})$ that
are diagonal with respect to this basis, have
$\lambda(h)\in\mathcal{L}_{\rho}$, and have $\ell^{\rm{H}}(h)\leq T$ is
compact. Since every $h$ in the compact family $\mathscr{C}_{\rho,T}$ acts by
diffeomorphisms on $\mathbb{R}\mathbb{P}^{n-1}$, there are uniform lower and
upper bounds on $D_{p}h(v)/||v||$ for $p\in\mathbb{R}\mathbb{P}^{n-1}$,
nonzero $v\in\text{T}_{p}\mathbb{R}\mathbb{P}^{n-1}$ and
$h\in\mathscr{C}_{\rho,T}$. ∎

Now let $T>0$. Since $K$ is compact, there is a $\nu>0$ so the lower bound
$\delta(\Phi_{\rho}(p),\Phi_{\rho}(q))>\nu$ holds for all $(p,q)\in K$. We
claim with $B(\rho,T)$ as supplied by Claim 4.10, $\tau=\nu/B(\rho,T)$
satisfies the condition we reduced to above. So suppose there is a pair
$(p,q)\in B_{n}^{(2)}$ so that
$\delta_{g}(\Phi_{\rho}(p),\Phi_{\rho}(q))<\nu/B(\rho,T)$ and $(\gamma
p,\gamma q)\in K$. Then
$\delta_{g}(\rho(\gamma)\Phi_{\rho}(p),\rho(\gamma)\Phi_{\rho}(q))>\nu$, so
the action of $\rho(\gamma)$ on $\mathbb{R}\mathbb{P}^{n-1}$ is not
$B(\rho,T)$-Lipschitz and Claim 4.10 shows $\ell^{\text{H}}_{\rho}(\gamma)>T$.
∎

####  4.2.4. Estimates Near the Frenet Curve

We now prove our lower bound for the Hölder exponent of $\Phi_{ab}$. Let

(4.2)
$\displaystyle\alpha_{ab}^{U}=\min\left(\alpha_{ab}^{L},\inf_{\gamma\in\Gamma-\\{e\\}}\frac{\ell_{\rho_{a}}^{\alpha_{1}}(\gamma)}{\ell_{\rho_{b}}^{\rm{H}}(\gamma)}\right)$

Note that $\alpha_{ab}^{U}>0$ as the limit cones $\mathcal{L}_{\rho_{a}}$ and
$\mathcal{L}_{\rho_{b}}$ are contained in $\mathring{\mathfrak{a}}^{+}$ and
$d_{\text{Th}}^{\alpha_{1}}(\rho_{b},\rho_{a})$ is finite.

######  Proposition 4.11.

Let $\rho_{a},\rho_{b}:\Gamma\to{\rm{PSL}}_{n}(\mathbb{R})$ ($n>3$) be
Hitchin. Then $\Phi_{ab}$ is $\alpha_{ab}^{U}$-Hölder.

###### Proof.

The idea is that points that are both close together in
$\mathcal{B}_{\rho_{c}}$ and close to $\Xi^{1}_{\rho_{c}}$ ($c\in\\{a,b\\}$)
are the image under $\Phi_{\rho_{c}}$ of pairs of points in $B_{n}$ specified
by a quadruple of clustered-together points in $\partial\Gamma$. The argument
of the Fuchsian case is effective for such points with the two modifications
of using loxodromy in place of proximality and the relevant application of the
uniform convergence group property of $\Gamma$ using the points on the outside
of the cluster. The need to control more quantities in this setting makes the
proof rather technically involved; the reader may wish to refer back to §2.1
for a model.

Our proof breaks into the following stages, which we have labelled below:

  1. (1)

Notation and basic features of pairs of points under consideration,

  2. (2)

Initial Abels-Margulis-Soifer [1] application and the main construction,

  3. (3)

Application of loxodromy refinement to control placements of involved points,

  4. (4)

Uniformly controlled choice of affine chart,

  5. (5)

The final computation.

(1). Let us set notation. Fix a background metric $d_{\partial\Gamma}$ on
$\partial\Gamma$. Writing
$B_{n}=(\partial\Gamma_{1}^{2}\sqcup...\sqcup\partial\Gamma_{k_{n}}^{2})/\sim$,
we distinguish between membership in each term by denoting a pair
$(x,y)\in\partial\Gamma_{j}^{2}$ ($j\in\mathcal{K}_{n}$) by $(x,y)_{j}$. For
any $\kappa,\eta>0$ and $c\in\\{a,b\\}$, we define
$\Xi^{2}_{c}(\kappa)\subset\Xi^{2}_{\rho_{c}}$ to be the complement of
$N_{\kappa}(\Xi^{1}_{\rho_{c}})$, and
$\text{Far}_{c}(\eta)\subset\mathcal{B}_{\rho_{c}}^{(2)}$ to be the points
$(l_{1},l_{2})$ so $\delta_{0}(l_{1},l_{2})\geq\eta$. Let $U_{\kappa,c}$ be
$N_{\kappa}(\Xi^{1}_{\rho_{c}})$ and
$V_{\eta,c}=\mathcal{B}_{\rho_{c}}^{(2)}-\text{Far}_{c}(\eta)$. Let
$\mathscr{O}_{\kappa}\subset B_{n}$ be
$\Phi_{a}^{-1}(U_{\kappa,a})\cap\Phi_{b}^{-1}(U_{\kappa,b})$. Let
$\mathscr{Q}_{\kappa,\eta}\subset B_{n}^{(2)}$ be the collection of pairs
$(p,q)$ with $p,q\in\mathscr{O}_{\kappa}$ and $\Phi_{c}(p,q)\in V_{\eta,c}$
for all $c\in\\{a,b\\}$.

Using Lemma 4.4, for any $\kappa$ and $\eta$, there are constants
$C_{\Xi}(\kappa)$ and $C_{\text{Far}}(\eta)$ so that

(4.3)
$\displaystyle\frac{\delta_{0}(\Phi_{\rho_{a}}(p),\Phi_{\rho_{a}}(q))}{\delta_{0}(\Phi_{\rho_{b}}(p),\Phi_{\rho_{b}}(q))^{\alpha_{ab}^{U}}}$
$\displaystyle\leq C_{\Xi}(\kappa)$
$\displaystyle(p,q\notin\mathscr{O}_{\kappa}),$
$\displaystyle\frac{\delta_{0}(\Phi_{\rho_{a}}(p),\Phi_{\rho_{a}}(q))}{\delta_{0}(\Phi_{\rho_{b}}(p),\Phi_{\rho_{b}}(q))^{\alpha_{ab}^{U}}}$
$\displaystyle\leq C_{\text{Far}}(\eta)$
$\displaystyle((\Phi_{\rho_{b}}(p),\Phi_{\rho_{b}}(q))\in\text{Far}_{b}(\eta)).$

We say points $x_{1},x_{2},x_{3},x_{4}\in\partial\Gamma$ are $\nu$-clustered
if $\text{max}_{1\leq i\neq j\leq 4}d_{\partial\Gamma}(x_{i},x_{j})<\nu.$ We
do not require the points be distinct. If $x_{1},x_{2},x_{3},x_{4}$ are
$\nu$-clustered, we call $\\{x_{1},x_{2},x_{3},x_{3}\\}$ a $\nu$-cluster. For
any $\nu$ sufficiently small, the complement of any $\nu$-clustered
$x_{1},x_{2},x_{3},x_{4}\in\partial\Gamma$ contains an interval $I$ with
diameter exceeding $2\text{diam}(\partial\Gamma)/3$. Then there are
$x_{l},x_{r}\in\\{x_{1},x_{2},x_{3},x_{4}\\}$ so that $\partial
I=\\{x_{l},x_{r}\\}$ and for any $z\in I$ the triple $(x_{l},x_{r},z)$ is
positively oriented. Call $x_{l},x_{r}$ the edge points of
$x_{1},x_{2},x_{3},x_{4}$.

We shall repeatedly take the cluster associated to a pair of points
$(p,q)\in\mathcal{Q}_{\eta,\kappa}$ by writing
$p=(x_{1},y_{1})_{i},q=(x_{2},y_{2})_{j}$ and considering
$\\{x_{1},y_{1},x_{2},y_{2}\\}$. Let us denote this cluster by
$\text{cl}(p,q)$. We note that some data is forgotten by taking
$\text{cl}(p,q)$: which copy of $\partial\Gamma^{2}_{k}$ points are in, and
the order of points in $\partial\Gamma^{2}_{k}$. This data is salient away
from $\Xi^{1}_{\rho_{c}}$ ($c\in\\{a,b\\}$), and does not end up needing to be
explicitly used in the following.

The following is a direct consequence of $\Phi_{\rho}$ being a homeomorphism:

######  Claim 4.12 (Clustering Characterization).

For any $\nu>0$ there are $\eta(\nu),\kappa(\nu)>0$ so that ${\rm{cl}}(p,q)$
is a $\nu$-cluster for all $(p,q)\in\mathscr{Q}_{\kappa,\eta}$.

On the other hand, given any $\nu>0$, there are strictly positive
$\kappa(\nu)$ and $\eta(\nu)$ so that if $(p,q)\in B_{n}^{(2)}$ has
${\rm{cl}}(p,q)$ not a $\nu$-cluster, then for each $c\in\\{a,b\\}$ one of the
following holds:

  1. (1)

Both $\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q)$ are in
$\Xi^{2}_{c}(\kappa(\nu))$,

  2. (2)

$(\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q))\in{\rm{Far}}_{c}(\eta(\nu))$.

###### Proof.

The first claim is clear. The second claim is clear, with the weakened
conclusion that one of
$\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q)\in\Xi^{2}_{c}(\kappa^{\prime}(\nu))$ or
$(\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q))\in\text{Far}_{c}(\eta^{\prime}(\nu))$.
We can then take $\eta^{\prime\prime}>0$ so that if
$\Phi_{\rho_{c}}(p)\in\Xi^{2}_{c}(\kappa^{\prime}(\nu))$ and
$\Phi_{\rho_{c}}(q)\in U_{\kappa^{\prime}(\nu)/2,c}$ then
$(\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q))\in\text{Far}_{c}(\eta^{\prime\prime})$.
Adjusting to $\kappa=\kappa^{\prime}(\nu)/2$ and
$\eta=\text{min}(\eta^{\prime},\eta^{\prime\prime})$ then gives the claim. ∎

Due to Lemma 4.4, equations (4.3), and Claim 4.12, for the conclusion we may
take arbitrarily small but fixed $\nu,\kappa,\eta>0$ to be determined later
and only consider points $(p,q)\in\mathscr{Q}_{\kappa,\eta}$ so that
$\text{cl}(p,q)$ is a $\nu$-cluster. We call points $(p,q)$ satisfying the
above $(\kappa,\eta,\nu)$-adapted, and write the collection of
$(\kappa,\eta,\nu)$-adapted pairs as $\mathscr{R}_{\kappa,\eta,\nu}$.

(2). The next piece of setup parallels §2.1 directly. Let $r>\varepsilon>0$ be
sufficiently small and fixed. By Fact 4.5, there is a finite set
$F\subset\Gamma$ so that for all $\gamma\in\Gamma$ there is an $\eta\in F$ so
that $\rho_{a}(\eta\gamma)$ and $\rho_{b}(\eta\gamma)$ are
$(r,\varepsilon)$-loxodromic. As $\Gamma$ is a uniform convergence group,
there is a compact subset $K\subset\partial\Gamma^{(3)+}$ so that for any
$t\in\partial\Gamma^{(3)+}$ there is a $\gamma\in\Gamma$ so that $\gamma t\in
K$. Replacing $K$ with the compact set $L=FK$ and setting

$\Gamma_{(r,\varepsilon)}^{a,b}=\\{\gamma\in\Gamma\mid\rho_{a}(\gamma)\text{
and }\rho_{b}(\gamma)\text{ are }(r,\varepsilon)\text{ loxodromic}\\},$

for every $t\in\partial\Gamma^{(3)+}$ there is a
$\gamma\in\Gamma_{(r,\varepsilon)}^{a,b}$ so $\gamma t\in L$.

We analyze the following construction for the rest of the proof. Let
$(p,q)\in\mathscr{R}_{\kappa_{0},\eta_{0},\nu_{0}}$ for initial
$\kappa_{0},\eta_{0},\nu_{0}>0$ with $\nu_{0}$ less than
$\text{diam}(\partial\Gamma)/3$. Then $\text{cl}(p,q)$ is a $\nu_{0}$-cluster
with at least two distinct points. Let $x_{l}\neq x_{r}$ be the edge points.
Pick any third point $z$ so
$d_{\partial\Gamma}(z,x)>\text{diam}(\partial\Gamma)/3$ for all
$x\in\text{cl}(p,q)$. Then there is some
$\gamma_{pq}\in\Gamma_{r,\varepsilon}^{a,b}$ so that
$\gamma_{pq}(x_{l},x_{r},z)\in L$.

(3). As in §2.1, our freedom to choose $\eta,\kappa,\nu$ small allows us to
only consider $\gamma_{pq}$ so that both $\gamma_{pq}$ and $\gamma_{pq}^{-1}$
are elements of $\Gamma^{a,b}_{(r,\varepsilon^{\prime})}$ for a fixed
$\varepsilon^{\prime}>0$ of our choosing:

######  Claim 4.13.

For any $\varepsilon^{\prime}>0$ there are $\nu,\kappa,\eta>0$ so that if
$(p,q)\in\mathscr{R}_{\kappa,\eta,\nu}$ then
$\gamma_{pq}\in\Gamma^{a,b}_{(r,\varepsilon^{\prime})}$ and
$\gamma_{pq}^{-1}\in\Gamma^{a,b}_{(r,\varepsilon^{\prime})}$. We call such
points $(\kappa,\eta,\nu,\varepsilon^{\prime})$-adapted, and write the
collection of $(\kappa,\eta,\nu,\varepsilon^{\prime})$-adapted pairs as
$\mathscr{S}_{\kappa,\eta,\nu,\varepsilon^{\prime}}$.

###### Proof.

We retain the above notation. As $L$ is compact, there is a $r_{1}>0$ so that
$d_{\partial\Gamma}(x_{i},x_{j})\geq r_{1}$ for all $(x_{1},x_{2},x_{3})\in L$
and $1\leq i\neq j\leq 3$. As $\gamma_{pq}(x_{l},x_{r},z)\in L$ and
$\text{cl}(p,q)$ is a set with at most four elements there is a $r_{2}>0$ so
that for each $c\in\\{a,b\\}$ either:

  1. (1)

$(\Phi_{\rho_{c}}(\gamma_{pq}p),\Phi_{\rho_{c}}(\gamma_{pq}q))\in\text{Far}_{c}(r_{2})$,

  2. (2)

At least one of $\Phi_{\rho_{c}}(\gamma_{pq}p),\Phi_{\rho_{c}}(\gamma_{pq}q)$
is not in $U_{r_{2},c}$.

For $c\in\\{a,b\\}$ let $K_{r_{2},c}\subset B_{n}^{2}$ be the compact set of
pairs $(p,q)$ so that $\delta_{0}(\Phi_{c}(p),\Phi_{c}(q))\geq r_{2}$.

In the first case $(p,q)\in\mathscr{R}_{\kappa,\eta,\nu}$ so
$\delta_{0}(\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q))\leq\eta$ and
$(\gamma_{pq}p,\gamma_{pq}q)\in K_{r_{2},c}$. In the second case, there is a
point $w\in\partial\Gamma$ and $p^{\prime}\in\\{p,q\\}$ so
$\delta_{0}(\Phi_{\rho_{c}}(p^{\prime}),\Phi_{\rho_{c}}(w,w))\leq\kappa$ and
$(\gamma_{pq}p^{\prime},\gamma_{pq}(w,w))\in K_{r_{2},c}$. We conclude from
the Loxodromy Refinement Proposition that for any $\varepsilon^{\prime}>0$
there is a $\tau(\varepsilon^{\prime})>0$ so that for any $\kappa,\eta,\nu$
with $\kappa,\eta<\tau(\varepsilon^{\prime})$ that $\rho_{c}(\gamma_{pq})$ is
$(r,\varepsilon^{\prime})$-biloxodromic for all $c\in\\{a,b\\}$, as desired. ∎

We next use the control over quantitative loxodromy given by Claim 4.13 to
arrange for $p,q,\gamma_{pq}p,\gamma_{pq}q,\gamma_{pq}^{+}$, and
$\gamma_{pq}^{-}$ to be sufficiently well-placed to give uniform control over
the choice of affine charts in the main estimate. The following consequence of
the hyperconvexity of $\xi_{\rho_{a}},\xi_{\rho_{b}}$ and that
$\Phi_{\rho_{c}}$ is a homeomorphism of compact metric spaces is useful in
this.

######  Claim 4.14.

For any $\varepsilon_{1}>0$ there is a $\tau>0$ so that for any
$c\in\\{a,b\\}$, $k\in\mathcal{K}_{n}$, and $z\in\partial\Gamma$:

  1. (1)

If $p=(x,y)_{k}$ and $\Phi_{\rho_{c}}(p)\in N_{\tau}(\xi^{n-1}_{\rho_{c}}(z))$
then $d_{\partial\Gamma}(x,z)<\varepsilon_{1}$ or
$d_{\partial\Gamma}(y,z)<\varepsilon_{1}$,

  2. (2)

If $p=(x,y)_{k}$ and $\Phi_{\rho_{c}}(p)\in N_{\tau}(\xi^{1}_{\rho_{c}}(z))$
then $d_{\partial\Gamma}(x,z)<\varepsilon_{1}$ and
$d_{\partial\Gamma}(y,z)<\varepsilon_{1}$.

###### Proof.

We prove the first claim. The second is similar. The space
${B}_{n}\times\partial\Gamma$ is compact, and the function
$\mu_{n}:B_{n}\times\partial\Gamma$ given by
$(p,z)\mapsto\delta_{0}(\Phi_{\rho_{c}}(p),\xi^{n-1}_{\rho_{c}}(z))$ is
continuous and zero exactly on the set $Z=\\{((x,y)_{k},z)\mid x=z\text{ or
}y=z\\}$, by the hyperconvexity of $\xi_{\rho_{c}}$.

The set $K_{\varepsilon}\subset B_{n}\times\partial\Gamma$ given by
$\\{((x,y)_{k},z)\mid d_{\partial\Gamma}(x,z)\geq\varepsilon_{1}\text{ and
}d_{\partial\Gamma}(y,z)\geq\varepsilon_{1}\\}$ is compact and disjoint from
$Z$. So the restriction of $\mu_{n}$ to $K_{\varepsilon_{1}}$ has a positive
infimum $\tau>0$. So any $p\in B_{n}$ with $\Phi_{\rho_{c}}(p)\in
N_{\tau}(\xi^{n-1}_{\rho_{c}}(z))$ has
$(p,z)\in(B_{n}\times\partial\Gamma)-K_{\varepsilon_{1}}$. This proves the
claim. ∎

We next constrain the placement of the edges of $\text{cl}(p,q)$ relative to
$\gamma_{pq}^{-}$ and the placement of $\gamma_{pq}z$ relative to
$\gamma_{pq}^{+}$. Recall $z=z(p,q)$ is the third point in $\partial\Gamma$
chosen in determining $\gamma_{pq}$.

######  Claim 4.15.

For all $\sigma^{+},\sigma^{-}>0$ there exist
$\eta,\kappa,\nu,\varepsilon^{\prime}>0$ so that for any
$(p,q)\in\mathscr{S}_{\eta,\kappa,\nu,\varepsilon^{\prime}}$, the edges
$x_{l},x_{r}$ of ${\rm{cl}}(p,q)$ satisfy:

  1. (1)

$d_{\partial\Gamma}(x_{l},\gamma_{pq}^{-}),d_{\partial\Gamma}(x_{r},\gamma_{pq}^{-})<\sigma^{-}$,

  2. (2)

$d_{\partial\Gamma}(\gamma_{pq}z,\gamma_{pq}^{+})<\sigma^{+}$.

###### Proof.

We begin with the first claim. We use the same dichotomy as in the proof of
Claim 4.13. First, let
$(\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q))\in\text{Far}_{c}(r_{2})$, so one of
$\Phi_{\rho_{c}}(\gamma_{pq}p)$ and $\Phi_{\rho_{c}}(\gamma_{pq}q)$ is not in
$N_{r_{2}/2}(\xi^{1}_{\rho_{c}}(\gamma_{pq}^{+}))$. If $\gamma_{pq}$ is
$(r,\varepsilon^{\prime})$-biloxodromic with $\varepsilon^{\prime}<r_{2}/2$,
this implies that at least one of $\Phi_{\rho_{c}}(p)$ and
$\Phi_{\rho_{c}}(q)$ is in
$N_{\varepsilon^{\prime}}(\xi_{\rho_{c}}^{n-1}(\gamma_{pq}^{-}))$. Without
loss of generality, take this to hold for $p$. Claim 4.14 then shows that for
$\varepsilon^{\prime}$ sufficiently small, writing $p=(x,y)_{k}$, some
$w\in\\{x,y\\}$ and $\gamma_{pq}^{-}$ have
$d_{\partial\Gamma}(w,\gamma_{pq}^{-})<\sigma^{-}/2$. With $\nu$ chosen so
$\nu<\sigma^{-}/2$, the claim holds.

The other case is that for at least one $p^{\prime}\in\\{p,q\\}$ there is a
$c\in\\{a,b\\}$ so $\Phi_{\rho_{c}}(\gamma_{pq}p^{\prime})$ is not in
$U_{r_{2},c}$. Then $\Phi_{\rho_{c}}(\gamma_{pq}p^{\prime})$ is not in
$N_{r_{2}}(\xi^{1}_{\rho_{c}}(\gamma_{pq}^{+}))$, and the claim follows
similarly.

We now arrange for the second conclusion to hold. By Claim 4.14, there is a
$\tau>0$ so that if
$\delta_{0}(\xi^{1}_{\rho_{c}}(\gamma_{pq}z),\xi^{1}_{\rho_{c}}(\gamma_{pq}^{+}))<\tau$
for some $c\in\\{a,b\\}$, then
$d_{\partial\Gamma}(\gamma_{pq}z,\gamma_{pq}^{+})<\sigma^{+}$. Since we have
arranged for $d_{\partial\Gamma}(z,x_{l})$ and $d_{\partial\Gamma}(z,x_{r})$
to be at least $\text{diam}(\partial\Gamma)/3$, there is a $r_{4}>0$
independent of $p,q$ so
$\delta_{0}(\xi^{1}_{\rho_{c}}(z),\Phi_{\rho_{c}}(p^{\prime}))\geq r_{4}$ for
all $p^{\prime}\in\\{p,q\\}$ and $c\in\\{a,b\\}$. Applying the first
conclusion with possibly decreased $\sigma^{-}$ gives constants
$(\kappa_{0},\eta_{0},\nu_{0},\varepsilon^{\prime}_{0})$ so that for
$(p,q)\in\mathscr{S}_{\kappa_{0},\eta_{0},\nu_{0},\varepsilon^{\prime}_{0}}$
the point $z$ is forced to be uniformly separated from $\gamma_{pq}^{-}$.
Applying Claim 4.14 produces a $r_{5}>0$ so the separation
$\delta_{0}(\Phi_{\rho_{c}}(z),\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{-}))\geq
r_{5}$ holds for all
$(p,q)\in\mathscr{S}_{\kappa_{0},\eta_{0},\nu_{0},\varepsilon^{\prime}_{0}}$
and $c\in\\{a,b\\}$. Now, take $\varepsilon^{\prime}$ so that
$\varepsilon^{\prime}<\min\\{\tau,\varepsilon^{\prime}_{0},r_{5}\\}$. Then
$\rho_{c}(\gamma_{pq})\xi^{1}_{\rho_{c}}(z)\in
N_{\tau}(\xi^{1}_{\rho_{c}}(\gamma_{pq}^{+}))$ for all
$(p,q)\in\mathscr{S}_{\kappa_{0},\eta_{0},\nu_{0},\varepsilon^{\prime}}$ and
$c\in\\{a,b\\}$. So
$d_{\partial\Gamma}(\gamma_{pq}z,\gamma_{pq}^{+})<\sigma^{+}$ for all such
$(p,q)$. ∎

The previous claim also allows us to separate both $p^{\prime}$ and
$\gamma_{pq}p^{\prime}$ from $\gamma_{pq}^{+}$ for $p^{\prime}\in\\{p,q\\}$.

######  Claim 4.16.

There is a $r_{6}>0$ so that for all sufficiently small
$(\kappa,\eta,\nu,\varepsilon^{\prime})>0$ and
$(p,q)\in\mathscr{S}_{\kappa,\eta,\nu,\varepsilon^{\prime}}$, for all
$c\in\\{a,b\\}$ and $p^{\prime}\in\\{p,q\\}$:

  1. (1)

$\delta_{0}(\Phi_{\rho_{c}}(p^{\prime}),\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{+}))\geq
r_{6}$,

  2. (2)

$\delta_{0}(\Phi_{\rho_{c}}(\gamma_{pq}p^{\prime}),\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{+}))\geq
r_{6}$.

###### Proof.

Corollary of Claims 4.14 and 4.15 together with the compactness of $L$. ∎

(4). We now define the affine charts in which we carry out our estimate. Let
$(p,q)\in\mathscr{S}_{\kappa,\eta,\nu,\varepsilon^{\prime}}$ be given for
small parameters to be specified later, and let $\gamma_{pq}$ be associated.
Let $x_{f}\in\\{x_{l},x_{r}\\}$ be the edge point of $\text{cl}(p,q)$ for
which $\gamma_{pq}x_{f}$ is as furthest from $\gamma_{pq}^{-}$. For
$c\in\\{a,b\\}$, take an affine chart $\mathcal{A}_{c}$ for
$\mathbb{R}\mathbb{P}^{n-1}$ so that in $\mathcal{A}_{c}$:

  1. (1)

$\xi^{1}_{\rho_{c}}(\gamma_{pq}^{-})$ is the origin,

  2. (2)

$\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{+})$ is the hyperplane at infinity,

  3. (3)

For a basis $v_{1},v_{2},...,v_{n-1}$ of
$\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{+})$ of eigenvectors of the restriction of
$\rho_{c}(\gamma_{pq})$ to $\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{+})$ listed in
order of decreasing modulus and with $v_{n}$ a nonzero vector in
$\xi^{1}_{\rho_{c}}(\gamma_{pq}^{-})$, the $x_{j}$ axis ($j=1,...,n-1$) is the
intersection of $\text{span}\\{v_{n},v_{j}\\}$ with $\mathcal{A}_{c}$,

  4. (4)

$\xi^{1}_{\rho_{c}}(\gamma_{pq}x_{f})=(1,...,1)$.

This specifies $\mathcal{A}_{c}$ completely. Note the final condition is
possible because of hyperconvexity of $\xi_{\rho_{c}}$, in particular [10]
Théorème 2 and Prop. 7. Let $\delta_{\mathcal{A}_{c}}$ be the Euclidean metric
on $\mathcal{A}_{c}$.

The following allows us to work in these well-adapted affine charts.

######  Claim 4.17.

There are $C,R<\infty$ and $\kappa,\eta,\nu,\varepsilon^{\prime}>0$ so that
for all $(p,q)\in\mathscr{S}_{\kappa,\eta,\nu,\varepsilon^{\prime}}$ for each
$c\in\\{a,b\\}$ and $p^{\prime}\in\\{p,q\\}$:

  1. (1)

$\Phi_{\rho_{c}}(p^{\prime})$ and $\Phi_{\rho_{c}}(\gamma_{pq}p^{\prime})$ are
in $N_{R}(0)$ in $\mathcal{A}_{c}$ (with respect to
$\delta_{\mathcal{A}_{c}}$),

  2. (2)

On $N_{R}(0)\subset\mathcal{A}_{c}$ the metrics $\delta_{\mathcal{A}_{c}}$ and
$\delta_{0}$ are $C$-multiplicatively comparable.

###### Proof.

The key fact is that $\text{PGL}_{n}(\mathbb{R})$ acts simply transitively on
$(n+1)$-ples of points in $\mathbb{R}\mathbb{P}^{n-1}$ in general position.
Denote by $l_{0},l_{1},...,l_{n}$ the points in $\mathbb{R}\mathbb{P}^{n-1}$
so that in the model inducing the metric $\delta_{0}$ with $e_{1},...,e_{n}$
an orthonormal basis for $\mathbb{R}^{n}$ we have $l_{i}=[e_{i}]$
($i=1,...,n$), and $l_{0}$ a fixed chosen point so $l_{0},l_{1},...,l_{n}$ are
in general position. Let $\mathcal{A}_{0}$ be the affine chart on
$\mathbb{R}\mathbb{P}^{n-1}$ specified by placing $l_{n}$ at the origin,
$\text{span}\\{l_{1},...,l_{n-1}\\}$ as the hyperplane at infinity, the
$x_{j}$-axis as $\text{span}\\{v_{n},v_{j}\\}$ ($j=1,...,n-1$) and $l_{0}$ at
$(1,...,1)$.

The affine chart $\mathcal{A}_{c}$ differs from $\mathcal{A}_{0}$ by the
element $g_{pq}$ of $\text{PGL}_{n}(\mathbb{R})$ that has $g[l_{i}]=v_{i}$ for
$i=1,...,n$ and $g[l_{0}]=\xi^{1}_{\rho_{c}}(\gamma_{pq}x_{f})$. To prove
multiplicative comparability of $\delta_{\mathcal{A}_{c}}$ and $\delta_{0}$ on
the complement of a neighborhood of fixed radius about
$\xi^{n-1}_{\rho_{c}}(\gamma_{pq}^{+})$, it thus suffices to place
$\xi^{1}_{\rho_{c}}(\gamma_{pq}x_{f}),[v_{1}],...,[v_{n}]$ inside of a fixed
compact subset of the collection of $(n+1)$-ples of points in
$\mathbb{R}\mathbb{P}^{n-1}$ in general position.

Let us first note that there is a $r_{7}>0$ independent of $p,q$ so that
$d_{\partial\Gamma}(\gamma_{pq}x_{f},\gamma_{pq}^{-})>r_{7}$,
$d_{\partial\Gamma}(\gamma_{pq}x_{f},\gamma_{pq}^{+})>r_{7}$, and
$d_{\partial\Gamma}(\gamma_{pq}^{-},\gamma_{pq}^{+})>r_{7}$. Note the set
$\mathscr{N}_{r_{7}}$ of $r_{7}$-separated triples of points in
$\partial\Gamma$ is compact. With the convention $\xi^{0}(x)=\\{0\\}$ for all
$x\in\partial\Gamma$ in the edge cases,
$(\xi^{k-1}(x)\oplus\xi^{n-k}(z))\cap\xi^{1}(y)=\\{0\\}$ for any
$(x,y,z)\in\partial\Gamma^{(3)+}$, $c\in\\{a,b\\}$, and $k\in\\{1,...,n\\}$ by
[10] Théorème 2 and Prop. 7. So there is a $r_{8}>0$ so that for all
$(p,q)\in\mathscr{S}_{\kappa,\eta,\nu,\varepsilon^{\prime}}$, with
$\kappa,\eta,\nu\,\varepsilon^{\prime}$ sufficiently small to apply Claim
4.16, $c\in\\{a,b\\}$, and $(n-1)$-ples of eigenvectors
$v_{1},...,v_{k-1},v_{k+1},...,v_{n}$ of $\rho_{c}(\gamma_{pq})$,

$\displaystyle\delta_{0}(\xi^{1}_{\rho_{c}}(\gamma_{pq}x_{f}),\text{span}(v_{1},...,v_{k-1},v_{k+1},...,v_{n}))\geq
r_{8}.$

This, together with Prop. 4.7 places the point
$(\xi^{1}_{\rho_{c}}(\gamma_{pq}x_{f}),[v_{1}],[v_{2}],...,[v_{n}])$ inside of
a fixed compact subset $\mathscr{K}$ of the space of $(n+1)$-ples of points in
general position in $\mathbb{R}\mathbb{P}^{n-1}$. So for every $r_{9}>0$ there
is a $C(r_{9})<\infty$ so that $\delta_{\mathcal{A}_{c}}$ and $\delta_{0}$ are
$C(r_{9})$-comparable on the complement of a $r_{9}$-neighborhood of
$\xi^{n-1}(\gamma_{pq}^{+})$.

Applying Claims 4.14 and 4.15 produces for any sufficiently small $r_{9}>0$
quality constants $\kappa,\eta,\nu,\varepsilon^{\prime}>0$ so that if
$(p,q)\in\mathscr{S}_{\kappa,\eta,\nu,\varepsilon^{\prime}}$ then
$p,q,\Phi_{\rho_{c}}(p),\Phi_{\rho_{c}}(q)$ are not in
$N_{r_{9}}(\xi_{\rho_{c}}^{n-1}(\gamma_{pq}^{+}))$ for all $c\in\\{a,b\\}$.
The claim follows. ∎

(5). The preliminaries to the main computation are now finished. In the affine
chart $\mathcal{A}_{c}$,

$\displaystyle\rho_{c}(\gamma_{pq}^{-1})(x_{1},x_{2},...,x_{n-1})$
$\displaystyle=\left(\frac{\lambda_{n}}{\lambda_{1}}x_{1},\frac{\lambda_{n}}{\lambda_{2}}x_{2},...,\frac{\lambda_{n}}{\lambda_{n-1}}x_{n-1}\right)=\begin{pmatrix}\exp(-\ell^{\text{H}}_{\rho_{c}}(\gamma_{pq}^{-1}))x_{1}\\\
\exp(-\ell^{\alpha_{1}+\alpha_{2}+...+\alpha_{n-2}}_{\rho_{c}}(\gamma_{pq}^{-1}))x_{2}\\\
\vdots\\\
\exp(-\ell^{\alpha_{1}}_{\rho_{c}}(\gamma_{pq}^{-1}))x_{n-1}\end{pmatrix}^{T}$

so that for all $v,w\in\mathcal{A}_{c}$,

(4.4) $\displaystyle
e^{-\ell^{\text{H}}_{\rho_{c}}(\gamma_{pq}^{-1})}d_{\mathcal{A}_{c}}(v,w)\leq
d_{\mathcal{A}_{c}}(\rho_{c}(\gamma_{pq}^{-1})v,\rho_{c}(\gamma_{pq}^{-1})w)\leq
e^{-\ell^{\alpha_{1}}_{\rho_{c}}(\gamma_{pq}^{-1})}d_{\mathcal{A}_{c}}(v,w).$

Now let $(p,q)$ be $(\nu,\eta,\kappa,\varepsilon^{\prime})$-adapted as in
Claim 4.17. By Claim 4.12 applied to $L$ there is a $r_{10}>0$ so that for all
such $(p,q)$ and $c\in\\{a,b\\}$ the pair
$(\Phi_{\rho_{c}}(\gamma_{pq}p),\Phi_{\rho_{c}}(\gamma_{pq}q))$ is in
$\Xi^{2}_{c}(r_{10})^{2}$ or $\text{Far}_{c}(r_{10})$. We then have

$\displaystyle\frac{\delta_{\mathcal{A}_{a}}(\Phi_{\rho_{a}}(p),\Phi_{\rho_{a}}(q))}{\delta_{\mathcal{A}_{b}}(\Phi_{\rho_{b}}(p),\Phi_{\rho_{b}}(q))^{\alpha_{ab}^{U}}}$
$\displaystyle=\frac{\delta_{\mathcal{A}_{a}}(\rho_{a}(\gamma_{pq}^{-1})\Phi_{\rho_{a}}(\gamma_{pq}p),\rho_{a}(\gamma_{pq}^{-1})\Phi_{\rho_{a}}(\gamma_{pq}q))}{\delta_{\mathcal{A}_{b}}(\rho_{b}(\gamma_{pq}^{-1})\Phi_{\rho_{b}}(\gamma_{pq}p),\rho_{b}(\gamma_{pq}^{-1})\Phi_{\rho_{b}}(\gamma_{pq}q))^{\alpha_{ab}^{U}}}$
$\displaystyle\leq\frac{e^{-\ell^{\alpha_{1}}_{\rho_{a}}(\gamma_{pq}^{-1})}}{e^{-\alpha_{ab}^{U}\ell^{\text{H}}_{\rho_{b}}(\gamma_{pq}^{-1})}}\frac{\delta_{\mathcal{A}_{a}}(\Phi_{\rho_{a}}(\gamma_{pq}p),\Phi_{\rho_{a}}(\gamma_{pq}q))}{\delta_{\mathcal{A}_{b}}(\Phi_{\rho_{b}}(\gamma_{pq}p),\Phi_{\rho_{b}}(\gamma_{pq}q))^{\alpha_{ab}^{U}}}$
$\displaystyle\leq\exp(\alpha_{ab}^{U}\ell^{\text{H}}_{\rho_{b}}(\gamma_{\rho_{pq}}^{-1})-\ell^{\alpha_{1}}_{\rho_{a}}(\gamma_{pq}^{-1}))\max(C_{\Xi}(r_{10}),C_{\text{Far}}(r_{10}))$
$\displaystyle\leq\max(C_{\Xi}(r_{10}),C_{\text{Far}}(r_{10})).$

So $\Phi_{ab}$ is $\alpha_{ab}^{U}$-Hölder on a neighborhood $U$ of
$\Xi^{1}_{\rho_{b}}$. Lemma 4.4 shows that $\Phi_{ab}$ is
$\alpha_{ab}^{U}$-Hölder on the complement of a much smaller neighborhood $V$
of $\Xi^{1}_{\rho_{b}}$ with closure contained in $U$. The Lebesgue number
lemma implies $\Phi_{ab}$ is $\alpha_{ab}^{U}$-Hölder. ∎

###  4.3. Completeness

The completeness properties of the bouquet coupling metric $D_{B}$ follow from
a consequence of the Collar Lemma for Hitchin representations:

######  Theorem 4.18 (Lee-Zhang [17], Corollary 1.4).

Let $\mathcal{C}=\\{\gamma_{1},...\gamma_{k}\\}$ be a collection of curves on
$S$ containing a pants decomposition and so that the complement of
$\mathcal{C}$ in $S$ is a union of disks. Then the map
${\rm{Hit}}_{n}(S)\to\mathbb{R}^{k}$ given by
$\rho\mapsto(\ell_{\rho}^{\rm{H}}(\gamma_{1}),...,\ell_{\rho}^{\rm{H}}(\gamma_{k}))$
is proper.

Theorems B and D follow from the identity
$D_{B}^{*}(\rho_{a},\rho_{b})=\max(D_{B}(\rho_{a},\rho_{b}),D_{B}(\rho_{b},\rho_{a}))$
and:

######  Proposition 4.19.

Let $R>0$ and $\rho_{0}\in{\rm{Hit}}_{n}(S)$ for $n>3$. Then the forward ball
$B_{R,f}(\rho_{0})=\\{\rho\in{\rm{Hit}}_{n}(S)\mid D_{B}(\rho_{0},\rho)\leq
R\\}$ is compact.

###### Proof.

Since $D_{B}$ induces the standard topology on $\text{Hit}_{n}(S)$, the
forward ball $B_{R,f}$ is closed. So it suffices to find a compact set
containing $B_{R,f}(\rho_{0})$. Let
$\mathcal{C}=\\{\gamma_{1},...,\gamma_{k}\\}$ be a set of curves satisfying
the hypotheses of Theorem 4.18. If $\rho\in B_{R,f}(\rho_{0})$ there is a $K$
so that $\ell^{\alpha_{j}}_{\rho}(\gamma)<K$ for all
$\gamma_{i}\in\mathcal{C}$ and $j\in\\{1,...,n-1\\}$ by Theorem C and
finiteness of $\mathcal{C}$. So $\ell^{\text{H}}_{\rho}(\gamma)\leq(n-1)K$ for
all $\gamma\in\mathcal{C}$ and $\rho\in B_{R,f}(\rho_{0})$. Theorem 4.18 shows
that the collection of Hitchin representations $\rho$ with
$\ell_{\rho}^{\text{H}}(\gamma)\leq(n-1)K$ for all $\gamma\in\mathcal{C}$ is
compact. ∎

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