# Basmajian’s identity over non-Archimedean local fields

Yan Mary He  Department of Mathematics
University of Oklahoma
Norman, OK 73069<EMAIL_ADDRESS>
###### Abstract.

Let $\Sigma$ be a connected compact oriented surface with boundary and
negative Euler characteristic. Let $k$ be a non-Archimedean local field. In
this paper, we prove Basmajian’s identity for projective Anosov
representations $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ using Berkovich
hyperbolic geometry.

##  1\. Introduction

If $\Sigma$ is a connected compact oriented hyperbolic surface with geodesic
boundary $\partial\Sigma$, an orthogeodesic $\gamma$ on $\Sigma$ is a properly
immersed geodesic arc that is perpendicular to $\partial\Sigma$ at both ends.
In [2], Basmajian proved the following identity relating the sum of hyperbolic
lengths of boundary geodesics to the lengths of orthogeodesics on $\Sigma$:

$\ell(\partial\Sigma)=\sum_{\gamma}2\log\coth^{2}(\ell(\gamma)/2)$

where the summation is taken over all orthogeodesics $\gamma$ on $\Sigma$.

A hyperbolic structure on $\Sigma$ can be identified with a discrete faithful
representation $\rho:\pi_{1}\Sigma\to\text{PSL}(2,\mathbb{R})$ acting on the
hyperbolic plane as Möbius transformations. For simplicity, we look for now
surfaces with one boundary component. Up to conjugation, we may assume that
$\rho([\partial\Sigma])$ stabilizes the positive imaginary axis $L$.
Orthogeodesics are in one-to-one correspondence with double cosets of
$\pi_{1}(\partial\Sigma)$ in $\pi_{1}\Sigma$ [9, 18]. For each nontrivial
double coset $\pi_{1}(\partial\Sigma)w\pi_{1}(\partial\Sigma)$, the hyperbolic
geodesic $w\cdot L$ corresponds to another boundary component of
$\widetilde{\Sigma}$, and the contribution to Basmajian’s identity from this
term is $\log[\infty,0,\rho(w)(\infty),\rho(w)(0)]$, where $[a,b,c,d]$ is the
cross ratio on $\mathbb{R}\mathbb{P}^{1}$. Hence Basmajian’s identity can be
equivalently written as

$\ell({\partial\Sigma})=\sum_{w}\log[\infty,0,\rho(w)(\infty),\rho(w)(0)]$

where the summation is taken over all the double cosets
$\pi_{1}(\partial\Sigma)w\pi_{1}(\partial\Sigma)$.

In this paper, instead of considering the Archimedean field $\mathbb{R}$, we
consider projective Anosov representations
$\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ where $(k,|\cdot|)$ is a non-
Archimedean local field with a non-trivial absolute value $|\cdot|$. Following
Pozzetti-Sambarino-Wienhard [16, Definition 3.1], we say that a homomorphism
$\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ is projective Anosov if there exist
positive constants $c,\mu$ such that for all $g\in\pi_{1}\Sigma$ we have

$\frac{\sigma_{2}}{\sigma_{1}}(\rho(g))\leq ce^{-\mu|g|_{w}}$

where $|g|_{w}$ is the word length of $g$ with respect to a fixed generating
set. The numbers $\sigma_{i}$ are the ratios of the semi-homothecy; see
Section 2.2 for details. The main theorem of this paper is the following
Basmajian-type identity for projective Anosov representations
$\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$.

######  Theorem 1.1.

Let $\Sigma$ be a connected compact oriented surface with boundary and
negative Euler characteristic. Let
$\alpha_{1},\ldots,\alpha_{m}\in\pi_{1}\Sigma$ represent the free homotopy
classes of boundaries $a_{1},\ldots,a_{m}$. Let $k$ be a non-Archimedean local
field. If $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ is a projective Anosov
representation, then we have

(1.1)
$\sum_{j=1}^{m}\ell_{\rho}(\alpha_{j})=\sum_{j,q=1}^{m}\sum_{w\in\mathcal{L}_{j,q}}\log_{v}\left|C_{\rho}(\alpha_{j}^{+},\alpha_{j}^{-};w\cdot\alpha_{q}^{+},w\cdot\alpha_{q}^{-})\right|$

where $\alpha_{j}^{+},\alpha_{j}^{-}\in\partial\pi_{1}\Sigma$ are the
attracting and repelling fixed points of $\alpha_{j}$, respectively.

Here $\ell_{\rho}$ and $C_{\rho}$ denote certain length and cross ratio
associated to $\rho$ respectively, and $\log_{v}$ is the logarithm function
with base $p>1$ being the cardinality of the residue field of $k$. The set
$\mathcal{L}_{j,q}$ is a set of double coset representatives associated to
boundary components $a_{j}$ and $a_{q}$. We note that if we put an ordering on
a symmetric generating set of $\pi_{1}\Sigma$, each double coset
$\pi_{1}(\partial\Sigma)w\pi_{1}(\partial\Sigma)$ has a unique representative
$w$; namely, the shortest in word length and the largest with respect to the
ordering.

Before we outline the strategy of the proof of the theorem, we mention related
work on identities in hyperbolic geometry, higher Teichmüller theory and super
Teichmüller theory. We note that all these identities are for representations
over Archimedean fields. For identities on hyperbolic manifolds, we refer the
reader to the survey paper [4] and references therein. The author extended
Basmajian’s identity to certain Schottky representations [9]. Vlamis-Yarmola
established Basmajian’s identity for Hitchin representations [18]. The author
established Basmajian’s identity for hyperconvex Anosov representations [10].
Fanoni-Pozzetti obtained Basmajian-type inequalities for maximal
representations [16]. Moreover, there is recent progress in generalizing
McShane’s identity to higher Teichmüller theory [15, 12] and super Teichmüller
theory [11].

###  1.1. Strategy of the proof

We prove Theorem 1.1 using Berkovich hyperbolic geometry. If $k$ is a non-
Archimedian local field which is complete with respect to a non-trivial
absolute value $|\cdot|$, then the topology on $k$ induced by the given
absolute value is Hausdorff but it is totally disconnected. Berkovich’s theory
allows us to overcome this difficulty coming from the topology of $k$ and
study analytic objects on $k$. The Berkovich affine line
$\mathbb{A}^{1,an}_{k}$ is a locally compact, Hausdorff and path-connected
topological space which contains $k$, with the topology induced by $|\cdot|$,
as a dense subspace. The Berkovich projective line $\mathbb{P}^{1,an}_{k}$ is
obtained by adding the point $\infty$ to compactify $\mathbb{A}^{1,an}_{k}$.
Therefore, the Berkovich projective line $\mathbb{P}^{1,an}_{k}$ is a compact,
Hausdorff and path-connected topological space which contains the projective
line $\mathbb{P}^{1}(k)$ as a dense subspace. Moreover,
$\mathbb{P}^{1,an}_{k}$ admits a tree structure and it is in fact an
$\mathbb{R}$-tree. Based on the tree structure, the boundary of
$\mathbb{P}_{k}^{1,an}$ is $\mathbb{P}^{1}(k)$ and the set
$\mathbb{H}_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{P}_{k}^{1,an}\setminus\mathbb{P}^{1}(k)$
is called the Berkovic hyperbolic space.

The group $\text{PGL}(2,k)$ of Möbius transformations acts continuously on
$\mathbb{P}^{1,an}_{k}$ in a natural way which is compatible with its action
on $\mathbb{P}^{1}(k)$. This action preserves the Berkovic hyperbolic space
$\mathbb{H}_{k}$, and moreover, there exists a metric $d_{\mathbb{H}}$ on
$\mathbb{H}_{k}$ such that $\text{PGL}(2,k)$ acts by isometries.

Let $\Sigma$ be a surface as in Theorem 1.1. A projective Anosov
representation $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ gives rise to an action
of $\rho(\pi_{1}\Sigma)$ on $\mathbb{H}_{k}$. Since
$\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ is projective Anosov, it is discrete
and faithful, and for every $g\in\pi_{1}\Sigma$, the matrix $\rho(g)$ is
hyperbolic as an element in $\text{PGL}(2,k)$. The limit set of $\rho$ is the
image of the Gromov boundary $\partial\pi_{1}\Sigma$ under the limit map
$\zeta^{1}_{\rho}:\partial\pi_{1}\Sigma\to\mathbb{P}^{1}(k)$.

It turns out that we can adapt Basmajian’s decomposition to prove Theorem 1.1.
More specifically, fixing a boundary element $\alpha_{j}$, we look at a
fundamental domain $\mathbb{T}$ of the axis $L$ of $\rho(\alpha_{j})$ in the
Berkovich hyperbolic space $\mathbb{H}_{k}$. We show that $\mathbb{T}$ can be
decomposed into a Cantor set of measure zero, plus a countable collection of
complementary intervals. The Cantor set corresponds to the projection of the
limit set of $\rho$ onto the axis $L$, which can be proven to have
(hyperbolic) measure zero. On the other hand, the endpoints of each interval
is the projection of $\zeta^{1}_{\rho}(x^{+})\in\mathbb{P}^{1}(k)$ and
$\zeta^{1}_{\rho}(x^{-})\in\mathbb{P}^{1}(k)$ onto $L$, where
$x^{\pm}\in\partial\pi_{1}\Sigma$ are some translate of the endspoints of a
boundary element $\alpha_{q}$. Each of these intervals corresponds to a double
coset
$\langle\alpha_{j}\rangle\backslash\pi_{1}\Sigma/\langle\alpha_{q}\rangle$ and
the $d_{\mathbb{H}}$-length of each interval is given by log of the
corresponding cross ratio. Moreover, the intervals are mutually disjoint.
Finally, the identity is obtained by summing over all the boundary components
$\alpha_{j}$.

###  1.2. Organization of the paper

The paper is organized as follows. In Section 2, we review and prove
preliminary results about Anosov representations over non-Archimedean local
fields. In Section 3, we review and prove preliminary results about the
Berkovich projective line and the Berkovich hyperbolic space. In Section 4, we
prove Theorem 1.1.

### Acknowledgement

The author would like to thank Charles Favre for organizing the arithmetic
dynamics seminar at MSRI in Spring 2022 and for being inspiring.

##  2\. Anosov representations over non-Archimedean local fields

In this section, we first give a brief overview of definitions and properties
of Anosov representations $\rho:\Gamma\to\text{PGL}(n,k)$ from a Gromov
hyperbolic group $\Gamma$ into $\text{PGL}(n,k)$ where $k$ is a non-
Archimedean local field. We then discuss a cross ratio on the Gromov boundary
of $\Gamma$ associated to a projective Anosov representation. Finally, if
$\Sigma$ is a surface with boundary, we show that any projective Anosov
representation $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ is the restriction to
$\pi_{1}\Sigma$ of a doubling representation
$\hat{\rho}:\pi_{1}\hat{\Sigma}\to\text{PSL}(2,k)$ where $\hat{\Sigma}$ is the
double of $\Sigma$.

###  2.1. Non-Archimedean local fields and Haar measures

In this subsection we collect the basics of non-Archimedean local fields and
Haar measures. Standard references are [5, 17].

Let $k$ be a field. An absolute value on $k$ is a map
$|\cdot|:K\to\mathbb{R}_{\geq 0}$ such that

$|x|=0\iff x=0;|xy|=|x|\cdot|y|,\forall x,y\in k;\text{ and
}|x+y|\leq|x|+|y|.$

An absolute value on $k$ is called non-Archimedean if it also satisfies the
ultrametric inequality

$|x+y|\leq\max\\{|x|,|y|\\}.$

Otherwise the absolute value is called Archimedean. The absolute value
$|\cdot|$ on $k$ induces a metric on $k$; namely $d(x,y)=|x-y|$ for any
$x,y\in k$.

A field is complete if every Cauchy sequence has a limit point in the field.
It is a classical theorem that if $k$ is complete with respect to an
Archimedean absolute value, then $k$ is isomorphic to $\mathbb{R}$ or
$\mathbb{C}$ and the absolute value is equivalent to the usual absolute value.

Let $k$ be a field with a non-trivial non-Archimedean absolute value
$|\cdot|$. Let $k^{\times}=k\setminus\\{0\\}$ and $|k^{\times}|$ the value
group of $k$. The set
$\mathcal{O}_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in
k:|x|\leq 1\\}$ is called the ring of integers of $k$. The open unit ball
$\mathfrak{m}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in
k:|x|<1\\}$ is the unique non-zero prime ideal in $\mathcal{O}_{k}$. Their
quotient $\mathcal{O}_{k}/\mathfrak{m}$ is the residue field of $k$. An
absolute value on $k$ is discrete if the value group $|k^{\times}|$ is
discrete. A non-Archimedean local field is a field which is complete with
respect to a nontrivial discrete non-Archimedean absolute value such that the
residue field is finite. If $k$ is a non-Archimedean local field, then
$\mathcal{O}_{k}$ is compact and therefore $k$ is locally compact with respect
to the topology induced by $|\cdot|$. A non-Archimedean local field is
isomorphic to either a finite extension of the $p$-adic numbers
$\mathbb{Q}_{p}$ if it has characteristic zero, or the field of formal Laurent
series $\mathbb{F}_{q}((T))$ over a finite field $\mathbb{F}_{q}$ where $q$ is
a power of a prime number $p$ if it has characteristic $p$.

If $X$ is a locally compact Hausdorff space, the $\sigma$-algebra
$\mathcal{F}$ of $X$ is the collection of subsets of $X$ generated by open and
closed sets under countable unions and countable intersections. A Radon
measure $\mu$ on $X$ is a countably additive function

$\mu:\mathcal{F}\to\mathbb{R}_{\geq 0}\cup\\{\infty\\}$

such that

  1. (1)

$\mu(A)<\infty$ if $A$ is compact,

  2. (2)

$\mu(A)=\inf\\{\mu(U):A\subset U,U\text{ open}\\}$,

  3. (3)

$\mu(A)=\sup\\{\mu(C):C\subset A,C\text{ compact}\\}$

for all Borel sets $A\in\mathcal{F}$.

A topological group is locally compact if it is locally compact and Hausdorff.
A (left) Haar measure $m$ on a locally compact group $G$ is a nonzero Radon
measure that is translation invariant, i.e.

$m(x+A)=m(A)$

for any $A\in\mathcal{F}$ and $x\in G$. The right Haar measure can be defined
analogously and it coincides with the left Haar measure if the group is
abelian.

######  Theorem 2.1 (Weil).

Every locally compact group $G$ has a Haar measure. Moreover, if $m$ and
$m^{\prime}$ are two Haar measures on $G$, then there exists $\lambda>0$ such
that $m(A)=\lambda m^{\prime}(A)$ for any Borel set $A$.

The additive group of a local field is a locally compact abelian topological
group and thus it has a Haar measure $m$. We can normalize $m$ so that
$m(\mathcal{O}_{k})=1$. The following lemma states that there exists a unique
absolute value on $k$ that is compatible with the Haar measure on $k$.

######  Lemma 2.2.

Let $k$ be a local field with discrete valuation $v$ and absolute value
$|\cdot|\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=p^{-v(\cdot)}$
where $p$ is the cardinality of the residue field. Let $m$ be a Haar measure
on $k$. Then for every $x\in k$ and any Borel set $A\subset k$ we have

$m(x\cdot A)=|x|m(A).$

###### Proof.

Let $c\in k\setminus\\{0\\}$. We note that $(f_{c}^{-1})_{*}m=m\circ f_{c}$ is
also a Haar measure on the additive group $k$ where $f_{c}(x)=cx$ is a
continuous automorphism of $k$. Since Haar measure is unique up to a constant,
it follows that

$(m\circ f_{c})(A)=m(cA)=|c|m(A)$

for any Borel set $A$. ∎

###  2.2. Projective Anosov representations

Let $\Gamma$ be a Gromov hyperbolic group. Let $(k,|\cdot|)$ be a non-
Archimedean local field. Let $n\geq 2$ be an integer and let $p$ be an integer
with $1\leq p\leq n-1$. Following Pozzetti-Sambarino-Wienhard [16, Definition
3.1] which generalizes the work of Bochi-Potrie-Sambarino [3], we say that a
homomorphism $\rho:\Gamma\to\text{PGL}(n,k)$ is $\\{a_{p}\\}$-Anosov if there
exist positive constants $c,\mu$ such that for all $\gamma\in\Gamma$ we have

$\frac{\sigma_{p+1}}{\sigma_{p}}(\rho(\gamma))\leq ce^{-\mu|\gamma|_{w}}$

where $|\gamma|_{w}$ is the word length of $\gamma$ with respect to a fixed
generating set. The numbers $\sigma_{i}$ are defined as follows. Let $V$ be a
$k$-vector space (e.g. $V=k^{n}$). Given a $k$-norm $||\cdot||_{V,k}$ on $V$,
we say that $g\in{\rm GL}(V,k)$ is a semi-homothecy if there exists a
$g$-invariant $k$-orthogonal decomposition $V=V_{1}\oplus...\oplus V_{m}$ and
$\sigma_{1},...,\sigma_{m}\in\mathbb{R}_{>0}$ such that for every $i=1,...,m$
and every $v_{i}\in V_{i}$ one has

$||gv_{i}||_{V,k}=\sigma_{i}||v_{i}||_{V,k}.$

The numbers $\sigma_{i}$ are called the ratios of the semi-homothecy. An
$\\{a_{1}\\}$-Anosov representation is called a projective Anosov
representation.

An important property of Anosov representations is that they admit equivariant
transverse limit maps given by the following proposition.

######  Proposition 2.3 ([16, Proposition 3.5]).

If $\rho:\Gamma\to\text{PGL}(n,k)$ is $\\{a_{p}\\}$-Anosov, then for any
geodesic ray $\\{\gamma_{n}\\}_{n\geq 1}$ with endpoint $x$, the limits

$\zeta^{p}_{\rho}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{n\to\infty}U_{p}(\rho(\gamma_{n}))$
$\zeta^{n-p}_{\rho}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lim_{n\to\infty}U_{n-p}(\rho(\gamma_{n}))$

exist and do not depend on the ray. Moreover, they define continuous
$\rho$-equivariant transverse maps
$\zeta^{p}:\partial\Gamma\to\mathcal{G}_{p}(k^{n})$ and
$\zeta^{n-p}:\partial\Gamma\to\mathcal{G}_{n-p}(k^{n})$, where
$\partial\Gamma$ is the Gromov boundary of $\Gamma$ and
$\mathcal{G}_{j}(k^{n})$ denotes the Grassmannian $j$-planes in $k^{n}$ for
$j=p,n-p$. Here $U_{p}(\rho(\gamma))$ denotes the Cartan attractor of
$\rho(\gamma)$ (see [16, p. 9]), i.e.

$U_{p}(\rho(\gamma))=k\cdot(e_{1}\oplus...\oplus e_{p})$

with $\rho(\gamma)=kak^{\prime}$ coming from the Cartan $KAK$ decomposition of
$GL(n,k)$ and $\\{e_{j}\\}_{j=1}^{n}$ being the standard basis for the vector
space $k^{n}$.

If $\rho:\Gamma\to\text{PGL}(n,k)$ is projective Anosov, then the set
$\zeta_{\rho}^{1}(\partial\Gamma)\subset\mathbb{P}(k^{n})$ is called the limit
set of $\rho$.

###  2.3. Cross ratios

Let $\Sigma$ be a connected compact oriented surface with boundary and
negative Euler characteristic. In this subsection, we define a cross ratio for
four points on the Gromov boundary $\partial\pi_{1}\Sigma$ of the hyperbolic
group $\pi_{1}\Sigma$ which is a natural generalization of Labourie’s cross
ratio; see [14, 15]. This is achieved by using the transverse limit maps of a
projective Anosov representation which take points of $\partial\pi_{1}\Sigma$
into the projective spaces, and then taking the cross ratio considered by
Falbel-Guilloux-Will [7] that we introduce now.

Let $k$ be any field. Let $n\geq 2$ be an integer and denote by $k^{n*}$ the
dual of the vector space $k^{n}$. Consider two non-empty sets
$\Omega\subset\mathbb{P}(k^{n})$ and $\Lambda\subset\mathbb{P}(k^{n*})$ such
that

$\varphi(\omega)\neq 0$

for any $\omega\in\Omega$ and $\varphi\in\Lambda$. In other words, $\Omega$ is
disjoint from all hyperplanes defined by points in $\Lambda$.

Let
$(\varphi,\varphi^{\prime},\omega,\omega^{\prime})\in\Lambda^{2}\times\Omega^{2}$
with $\omega\neq\omega^{\prime}$ so that $\omega$ and $\omega^{\prime}$ span a
projective line $(\omega\omega^{\prime})$. In [7], Falbel-Guilloux-Will
considered the following cross ratio

$[[\varphi,\varphi^{\prime},\omega,\omega^{\prime}]]\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[\varphi_{\omega,\omega^{\prime}},\varphi^{\prime}_{\omega,\omega^{\prime}},\omega,\omega^{\prime}]$

where $\varphi_{\omega,\omega^{\prime}}$ and
$\varphi^{\prime}_{\omega,\omega^{\prime}}$ are the points
$\text{ker}(\varphi)\cap(\omega\omega^{\prime})$ and
$\text{ker}(\varphi^{\prime})\cap(\omega\omega^{\prime})$ in
$(\omega\omega^{\prime})$ respectively, and $[a,b,c,d]$ is the usual cross
ratio for four distinct points in $\mathbb{P}^{1}(k)=(\omega\omega^{\prime})$,
namely $[a,b,c,d]=(a-c)(b-d)/(a-d)(b-c)$.

The cross ratio satisfies the following properties.

######  Lemma 2.4.

Given
$(\varphi,\varphi^{\prime},\omega,\omega^{\prime})\in\Lambda^{2}\times\Omega^{2}$,
the cross ratio $[[\varphi,\varphi^{\prime},\omega,\omega^{\prime}]]$
satisfies

(2.1)
$[[\varphi,\varphi^{\prime},\omega,\omega^{\prime}]]=\frac{\bar{\varphi}(\bar{\omega})\bar{\varphi}^{\prime}(\bar{\omega}^{\prime})}{\bar{\varphi}(\bar{\omega}^{\prime})\bar{\varphi}^{\prime}(\bar{\omega})}$

where $\bar{\varphi},\bar{\varphi}^{\prime},\bar{\omega}$ and
$\bar{\omega^{\prime}}$ are any lifts of $\varphi,\varphi^{\prime},\omega$ and
$\omega^{\prime}$. Moreover, if $\omega,\omega^{\prime},\omega^{\prime\prime}$
are three points in $\Omega$ and
$\varphi,\varphi^{\prime},\varphi^{\prime\prime}$ are three points in
$\Lambda$, we have

  1. (1)

$[[\varphi,\varphi^{\prime},\omega,\omega^{\prime}]]=1$ if and only if
$\varphi=\varphi^{\prime}$ or $\omega=\omega^{\prime}$,

  2. (2)

$[[\varphi,\varphi^{\prime},\omega,\omega^{\prime}]]=[[\varphi,\varphi^{\prime},\omega,\omega^{\prime\prime}]]\cdot[[\varphi,\varphi^{\prime},\omega^{\prime\prime},\omega^{\prime}]]$,
and

  3. (3)

$[[\varphi,\varphi^{\prime},\omega,\omega^{\prime}]]=[[\varphi,\varphi^{\prime},\omega^{\prime},\omega]]^{-1}$.

###### Proof.

We give a sketch of the proof here and refer the interested reader to [7,
Lemma 2.3, Proposition 2.4]. Equation (2.1) is obvious if
$\varphi=\varphi^{\prime}$. Suppose $\varphi\neq\varphi^{\prime}$, we choose a
basis $\\{e_{m}\\}_{m=1}^{n}$ of the vector space $k^{n}$ such that the lifts
$\bar{\varphi},\bar{\varphi}^{\prime},\bar{\varphi}_{\omega,\omega^{\prime}}$
and $\bar{\varphi}^{\prime}_{\omega,\omega^{\prime}}$ (in $k^{n*}$ and
$k^{n}$) are as follows:

$\bar{\varphi}=e_{1}^{*},\bar{\varphi}^{\prime}=e_{2}^{*},\bar{\varphi}_{\omega,\omega^{\prime}}=e_{2}+\bar{u},\bar{\varphi}^{\prime}_{\omega,\omega^{\prime}}=e_{1}+\bar{u}^{\prime}$

where $\bar{u}$ and $\bar{u}^{\prime}$ are vectors in
Span$(e_{3},\cdots,e_{n})$. Note that $\bar{\varphi}_{\omega,\omega^{\prime}}$
and $\bar{\varphi}^{\prime}_{\omega,\omega^{\prime}}$ form a basis for the
plane spanned by $\bar{\omega}$ and $\bar{\omega}^{\prime}$. Therefore
$\bar{\omega}$ and $\bar{\omega}^{\prime}$ can be written as

$\bar{\omega}=\lambda^{\prime}\bar{\varphi}^{\prime}_{\omega,\omega^{\prime}}+\lambda\bar{\varphi}_{\omega,\omega^{\prime}}\text{
and
}\bar{\omega}^{\prime}=\mu^{\prime}\bar{\varphi}^{\prime}_{\omega,\omega^{\prime}}+\mu\bar{\varphi}_{\omega,\omega^{\prime}}.$

With respect to the coordinate basis
$\\{\bar{\varphi}^{\prime}_{\omega,\omega^{\prime}},\bar{\varphi}_{\omega,\omega^{\prime}}\\}$,
in the projective line $(\omega\omega^{\prime})$,
$\varphi_{\omega,\omega^{\prime}}$ has projective coordinate $[1,\infty=1/0]$,
$\varphi^{\prime}_{\omega,\omega^{\prime}}=[1,0]$,
$\omega=[\lambda^{\prime},\lambda]=[1;\lambda/\lambda^{\prime}]$ and
$\omega^{\prime}=[\mu^{\prime},\mu]=[1;\mu/\mu^{\prime}]$.

Therefore, we have

$\frac{\bar{\varphi}(\bar{\omega})\bar{\varphi}^{\prime}(\bar{\omega}^{\prime})}{\bar{\varphi}(\bar{\omega}^{\prime})\bar{\varphi}^{\prime}(\bar{\omega})}=\frac{\mu\lambda^{\prime}}{\mu^{\prime}\lambda}=[\varphi_{\omega,\omega^{\prime}},\varphi^{\prime}_{\omega,\omega^{\prime}},\omega,\omega^{\prime}]$

proving equation (2.1).

Identities (1)-(3) can be verified by straightforward computations. ∎

Now we define a cross ratio for four points on the boundary
$\partial\pi_{1}\Sigma$. Let $\rho:\pi_{1}\Sigma\to\text{PGL}(n,k)$ be a
projective Anosov representation with limit maps $\zeta^{1}$ and
$\zeta^{n-1}$. We denote by $(\partial\pi_{1}\Sigma)^{4*}$ the set

$(\partial\pi_{1}\Sigma)^{4*}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{(x,y,u,v)\in(\partial\pi_{1}\Sigma)^{4}:x\neq
v,y\neq u\\}.$

The cross ratio is a continuous function
$C_{\rho}:(\partial\pi_{1}\Sigma)^{4*}\to\mathbb{R}$ given by

$C_{\rho}(x,y,u,v)=[[\zeta^{n-1}(x),\zeta^{n-1}(y),\zeta^{1}(u),\zeta^{1}(v)]].$

We note that if $\rho:\pi_{1}\Sigma\to\text{PSL}(n,\mathbb{R})$ is Hitchin,
then $C_{\rho}$ is Labourie’s cross ratio.

######  Lemma 2.5.

Let $\rho:\pi_{1}\Sigma\to\text{PGL}(n,k)$ be a projective Anosov
representation. The cross ratio $C_{\rho}$ satisfies the following properties

  1. (1)

$C_{\rho}(x,y,u,v)=0\iff x=u$ or $y=v$.

  2. (2)

$C_{\rho}(x,y,u,v)=1\iff x=y$ or $u=v$.

  3. (3)

$C_{\rho}(x,y,u,v)=C_{\rho}(x,y,u,v^{\prime})C_{\rho}(x,y,v^{\prime},v)$

  4. (4)

$C_{\rho}(x,y,u,v)=C_{\rho}(x,y,v,u)^{-1}$

###### Proof.

(1)-(4) can be checked directly by definition and Lemma 2.4. ∎

Given a projective Anosov representation
$\rho:\pi_{1}\Sigma\to\text{PGL}(n,k)$, the translation length of any
nontrivial element $\gamma\in\pi_{1}\Sigma$ is defined as

$\ell_{\rho}(\gamma)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\log_{v}\left|\frac{\lambda_{1}(\rho(\gamma))}{\lambda_{n}(\rho(\gamma))}\right|$

where $\lambda_{1}(\rho(\gamma))$ and $\lambda_{n}(\rho(\gamma))$ are the
eigenvalues of maximum and minimum absolute value of $\rho(\gamma)$,
respectively. The period of a nontrivial element $\gamma\in\pi_{1}\Sigma$ is
defined as

$P_{\rho}(\gamma)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\log_{v}\left|C_{\rho}(\gamma^{+},\gamma^{-},x,\gamma
x)\right|$

where $x$ is any point in
$\partial\pi_{1}\hat{\Sigma}\setminus\\{\gamma^{+},\gamma^{-}\\}$.

If $k=\mathbb{R}$ or $\mathbb{C}$, then translation lengths and periods become
the usual notion of (complex) lengths and (complex) periods. Moreover, similar
to the case $k=\mathbb{R}$ proved by Labourie [14, Proposition 5.8], the
period of a nontrivial element equals its translation length.

######  Lemma 2.6.

For any nontrivial element $\gamma\in\pi_{1}\Sigma$,
$P_{\rho}(\gamma)=\ell_{\rho}(\gamma)$.

###### Proof.

The proof follows from the same calculation which is carried in details in
[14, Proposition 5.8]. ∎

###  2.4. Doubling the representation

Let $\Sigma$ be a connected compact oriented surface with boundary whose
double $\hat{\Sigma}$ has genus at least $2$. The boundary of the surface is
oriented so that the surface is on the left of each boundary curve. Recall
that the double $\hat{\Sigma}$ of a connected surface with boundary $\Sigma$
is the surface obtained by identifying two copies of $\Sigma$ along the
boundary. The goal of this subsection is to prove the following proposition.

######  Proposition 2.7.

Let $\Sigma$ be a connected compact oriented surface with boundary whose
double $\hat{\Sigma}$ has genus at least $2$. Let $(k,|\cdot|)$ be a non-
Archimedean local field. Then any projective Anosov representation
$\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ is the restriction to $\pi_{1}\Sigma$
of some projective Anosov representation
$\hat{\rho}:\pi_{1}\hat{\Sigma}\to\text{PSL}(2,k)$.

The proof follows the same procedure as [15, Section 9]. We give a brief
sketch for completeness. We start with some definitions. A good homomorphism
based at a point $v$ in a boundary component $\partial_{v}$ is a homomorphism
$R:\pi_{1}(\Sigma,v)\to\text{PSL}(2,k)$ such that

  1. (1)

the matrix $R(\partial_{v})$ is diagonal with entries decreasing in absolute
value; and

  2. (2)

the image of all boundary components are hyperbolic; namely, the trace of the
matrix is strictly greater than 1 in absolute value.

Let $j_{0}:\Sigma\to\hat{\Sigma}$ and $j_{1}:\overline{\Sigma}\to\hat{\Sigma}$
be the inclusion maps. Here $\overline{\Sigma}$ is homeomorphic to $\Sigma$ so
that $\hat{\Sigma}=\Sigma\cup_{\partial\Sigma}\overline{\Sigma}$. Let

$J\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}.$

A $J$-extension of a good homomorphism $R$ is a homomorphism
$\hat{R}:\pi_{1}\hat{\Sigma}\to\text{PSL}(2,k)$ such that

  1. (1)

$\hat{R}\circ(j_{0})_{*}=R$.

  2. (2)

For any $\gamma\in\pi_{1}\hat{\Sigma}$, we have
$\hat{R}(\overline{\gamma})=J\hat{R}(\gamma)J$. Here
$\overline{\gamma}=j_{1}^{-1}\circ j_{0}(\gamma)$.

  3. (3)

For every arc $c$ in $\Sigma$ joining $v$ to a boundary component of $\Sigma$,
we have $\hat{R}(c*\bar{c}^{-1})=K_{c}JK_{c}^{-1}J$ for some
$K_{c}\in\text{PSL}(2,k)$.

######  Lemma 2.8.

Any good homomorphism admits a unique $J$-extension.

###### Proof.

The proof essentially follows the same argument as the proof of [15,
Proposition 9.2.1.2]. We give a brief sketch for completeness. Let
$a_{1},...,a_{m}$ be the boundary components of $\Sigma$ and $v\in a_{1}$. Let
$c_{1},...,c_{m}$ be arcs joining $v$ to $a_{1},...,a_{m}$. For $i=1,...,m$,
let $\beta_{i}\in\pi_{1}\Sigma$ be $\beta_{i}=[c_{i}]*[a_{i}]*[c_{i}]^{-1}$.
Define $G$ to be the following free product

$G=\pi_{1}\Sigma*\pi_{1}\Sigma*F_{m}$

where $F_{m}$ is a free group generated by $\\{x_{1},...,x_{m}\\}$.

Let $i_{0}:\pi_{1}\Sigma\to G$ and $i_{1}:\pi_{1}\Sigma\to G$ be the injection
of $\pi_{1}\Sigma$ to the first and the second factor of $G$ respectively.
Define $H$ to be the normal subgroup of $G$ generated by
$i_{0}(a_{1})^{-1}i_{1}(a_{1})$ and

$i_{0}(\beta_{j})^{-1}\cdot x_{j}\cdot i_{1}(\beta_{j})\cdot x_{j}^{-1}$

where $j=1,...,m$. Let

$\Gamma=G/H$

and we claim that $\Gamma\simeq\pi_{1}\hat{\Sigma}$. To see this, it suffices
to note that $x_{j}$ corresponds via the isomorphism to
$[c_{j}]*[\overline{c}_{j}]^{-1}$ for $j=1,...,m$.

Define $\hat{R}:G\to\text{PSL}(2,k)$ by $\hat{R}\circ i_{0}=R$, $\hat{R}\circ
i_{1}=JRJ$ and $\hat{R}(x_{j})=K_{c_{j}}JK_{c_{j}}^{-1}J$. Straightforward
computations show that $\hat{R}$ vanishes on $H$. Therefore $\hat{R}$ descends
to $G/H$ and abusing notation, we denote it again by $\hat{R}$ which is a
desired $J$-extension of $R$.

The uniqueness of $\hat{R}$ follows from the fact that $\pi_{1}\hat{\Sigma}$
is generated by $(j_{0})_{*}\pi_{1}\Sigma$, $(j_{1})_{*}\pi_{1}\Sigma$ and
$[c_{j}]*[\overline{c}_{j}]^{-1}$ for $j=2,...,m$. Any two $J$-extensions
would agree on the generators (up to conjugation). ∎

Let $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ be projective Anosov. A
$J$-extension of $\rho$ is a representation
$\hat{\rho}:\pi_{1}\hat{\Sigma}\to\text{PSL}(2,k)$ such that the $J$-extension
of any good representative of $\rho$ belongs to the same conjugacy class of
$\hat{\rho}$.

######  Lemma 2.9.

Let $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ be projective Anosov. Then there
exists a unique (up to conjugation) $J$-extension $\hat{\rho}$ of $\rho$.
Moreover, $\hat{\rho}$ is projective Anosov.

###### Proof.

The uniqueness up to conjugation follows from the uniqueness of $J$-extensions
given by Lemma 2.8. To prove the existence, it amounts to show that the
$J$-extension $\hat{R}$ of a good representative $R$ based at $v$ is
independent (up to conjugation) of the choice of a good representative based
at $v$ and the choice of $v$. This part follows the proof of [15, Theorem
9.2.2.2]. To see $\hat{\rho}$ is projective Anosov, we note that the
$J$-extension $\hat{R}$ of a good representative $R$ is projective Anosov by
construction given in the proof of the previous lemma. ∎

###### Proof of Proposition 2.7.

Proposition 2.7 follows from Lemma 2.9. ∎

##  3\. The Berkovich projective line $\mathbb{P}^{1,{\rm an}}_{k}$

Let $(k,|\cdot|)$ be a complete metrized non-Archimedean field. In this
section, we give a brief overview of the Berkovic projective line
$\mathbb{P}^{1,{\rm an}}_{k}$, the Berkovic hyperbolic space $\mathbb{H}_{k}$
and the group $\text{PGL}(2,k)$ that we will use in the next section. Our
exposition follows closely from [6] and [1].

###  3.1. The Berkovich projective line and the Berkovich hyperbolic space

Let $(k,|\cdot|)$ be a complete metrized non-Archimedean field. Recall that
$k^{\times}=k\setminus\\{0\\}$ and $|k^{\times}|$ is the value group of $k$.
Let $\mathbb{P}^{1}_{k}$ be the projective line over $k$, viewed as an
algebraic variety with Zariski toplogy, and by $\mathbb{P}^{1}(k)$ the set of
$k$-points of $\mathbb{P}^{1}_{k}$ which as a set is $k\cup\\{\infty\\}$. The
Berkovich analytification $\mathbb{P}^{1,{\rm an}}_{k}$ of
$\mathbb{P}^{1}_{k}$ is defined as follows. Recall that a multiplicative semi-
norm on a ring $A$ is a function $[~{}]_{x}:A\to\mathbb{R}_{\geq 0}$ such that

$[0]_{x}=0,[1]_{x}=1,[fg]_{x}=[f]_{x}\cdot[g]_{x}\text{ and
}[f+g]_{x}\leq[f]_{x}+[g]_{x}$

for all $f,g\in A$. We first define the Berkovich analytification
$\mathbb{A}^{1,{\rm an}}_{k}$ of the affine line $\mathbb{A}^{1}_{k}$ as the
set of multiplicative semi-norms on $k[Z]$ whose restriction to $k$ coincides
with $|\cdot|$, endowed with the topology of pointwise convergence. More
specifically, given a point $x\in\mathbb{A}^{1,{\rm an}}_{k}$ and a polynomial
$P\in k[Z]$, the value of the semi-norm defined by $x$ on $P$ is denoted by
$|P(x)|\in\mathbb{R}_{\geq 0}.$ The Gauss norm $\sum_{i}a_{i}Z^{i}\mapsto{\rm
max}|a_{i}|$ defines a point $x_{g}$ in $\mathbb{A}^{1,{\rm an}}_{k}$ which is
called the Gauss point. The Berkovich projective line $\mathbb{P}^{1,{\rm
an}}_{k}$ can be defined topologically as the one-point compactification of
$\mathbb{A}^{1,{\rm an}}_{k}$; namely, we have $\mathbb{P}^{1,{\rm
an}}_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{A}^{1,{\rm
an}}_{k}\cup\\{\infty\\}$.

Points in $\mathbb{P}^{1,{\rm an}}_{k}$ correspond to balls in
$\mathbb{P}^{1}(k)$. Recall that a closed (resp. open) ball in $k$ of radius
$R\geq 0$ is defined as

$\overline{B}(z_{0},R)=\\{z\in k:|z-z_{0}|\leq R\\}$

(resp. $B(z_{0},R)=\\{z\in k:|z-z_{0}|<R\\}$). A ball in $\mathbb{P}^{1}(k)$
is either a ball in $k$ or the complement of a ball in $k$. Any closed (or
open) ball $B\subset\mathbb{P}^{1}(k)$ determines a point
$x_{B}\in\mathbb{P}^{1,{\rm an}}_{k}$. Indeed, if $B$ or its complement is a
singleton $\\{z\\}$, then the point $x_{B}$ is the rigid point associated to
$z$. If $B$ is a (closed or open) ball of finite radius in $k$, then $x_{B}$
is the point in $\mathbb{A}^{1,{\rm an}}_{k}$ corresponding to the semi-norm

$|P(x_{B})|\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup_{B}|P|.$

If the complement of $B$ is a ball of finite radius, then we set

$|P(x_{B})|\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup_{k\setminus
B}|P|.$

The Gauss point $x_{g}$ corresponds to $x_{\bar{B}(0,1)}$. Moreover, the
Berkovich projective line $\mathbb{P}^{1,{\rm an}}_{k}$ is a (profinite)
$\mathbb{R}$-tree in the sense that it is uniquely path connected; namely, for
any two points $x,y\in\mathbb{P}^{1,{\rm an}}_{k}$, there is a unique segment
$[x,y]\subset\mathbb{P}^{1,{\rm an}}_{k}$.

A point in $\mathbb{P}^{1,{\rm an}}_{k}$ is said to be rigid if the
multiplicative semi-norm has a non-trivial kernel. There is a canonical
bijection between the set of closed points of the $k$-scheme
$\mathbb{P}^{1}_{k}$ and the set of rigid points in $\mathbb{P}^{1,{\rm
an}}_{k}$. To see this, let $z$ be a point in a finite extension of $k$. Then
the semi-norm $|\cdot|_{z}$ defined by $P\mapsto|P(z)|$ is a rigid point in
$\mathbb{A}^{1,{\rm an}}_{k}$. Therefore, $\mathbb{P}^{1}(k)$ naturally embeds
in $\mathbb{P}^{1,{\rm an}}_{k}$ as a set of rigid points.

Now suppose $(k,|\cdot|)$ is a complete metrized non-Archimedean field which
is algebraically closed. Points in $\mathbb{P}^{1,{\rm an}}_{k}$ can be
classified by their types. Points in $\mathbb{P}^{1,{\rm an}}_{k}$
corresponding to balls of diameter zero in $k$ are called type I points of
$\mathbb{P}^{1,{\rm an}}_{k}$. These points are in bijection with rigid points
in the Berkovich projective space $\mathbb{P}^{1,{\rm an}}_{k}$. Points in
$\mathbb{P}^{1,{\rm an}}_{k}$ corresponding to balls of diameter ${\rm
diam}(B)\in|k^{\times}|$ are said to be of type II. Points in
$\mathbb{P}^{1,{\rm an}}_{k}$ corresponding to balls of diameter ${\rm
diam}(B)\notin|k^{\times}|$ are said to be of type III.

More generally, points in $\mathbb{P}^{1,{\rm an}}_{k}$ are in bijection with
(equivalence classes of) decreasing sequences of balls. If $k$ is spherically
complete (i.e. every decreasing interseciton of balls is non-empty), then
$\mathbb{P}^{1,{\rm an}}_{k}$ consists of type I, II or III points. A local
field is spherically complete. In general, $\mathbb{P}^{1,{\rm an}}_{k}$ could
contain type IV points corresponding to decreasing sequences of balls whose
intersection is empty. For each $i=1,...,4$, the set of type $i$ points, if it
is non-empty, is dense in $\mathbb{P}^{1,{\rm an}}_{k}$.

Types of points in $\mathbb{P}^{1,{\rm an}}_{k}$ are related to the
$\mathbb{R}$-tree structure of $\mathbb{P}^{1,{\rm an}}_{k}$ as follows. Type
I and IV points are vertices of the $\mathbb{R}$-tree $\mathbb{P}^{1,{\rm
an}}_{k}$ which have only one branch; namely, they are the end points of the
tree. Type II points are vertices of $\mathbb{P}^{1,{\rm an}}_{k}$ which have
at least three branches. Finally type III points are the vertices of
$\mathbb{P}^{1,{\rm an}}_{k}$ which have exactly two branches; that is, the
regular points.

As a set, the Berkovich hyperbolic space $\mathbb{H}_{k}$ is defined as

$\mathbb{H}_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathbb{P}^{1,{\rm
an}}_{k}\setminus\mathbb{P}^{1}(k).$

It is a proper subtree of $\mathbb{P}^{1,{\rm
an}}_{k}\setminus\mathbb{P}^{1}(k)$ which is neither open nor closed.
Moreover, it contains no rigid points.

There is a partial order on $\mathbb{A}^{1,{\rm an}}_{k}$. If
$x,y\in\mathbb{A}^{1,{\rm an}}_{k}$, we say that $x\leq y$ if

$[f]_{x}\leq[f]_{y}\text{ for all }f\in k[Z].$

In terms of balls, if $x,y$ are of type I, II or III, we say that $x\leq y$ if
the ball corresponding to $x$ is contained in the ball corresponding to $y$.
For any pair of points $x,y\in\mathbb{A}^{1,{\rm an}}_{k}$, there exists a
unique least upper bound $x\cup y\in\mathbb{A}^{1,{\rm an}}_{k}$ with respect
to this partial order.

We now define a metric $d_{\mathbb{H}}$ on $\mathbb{H}_{k}$. If
$x,y\in\mathbb{H}_{k}$ with $x\leq y$, then we define

$d_{\mathbb{H}}(x,y)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\log_{v}\frac{{\rm
diam}(y)}{{\rm diam}(x)}$

where $\log_{v}$ is the logarithm function with base $p>1$ being the
cardinality of the residue field of $k$. More generally, if
$x,y\in\mathbb{H}_{k}$, we define

$d_{\mathbb{H}}(x,y)=d_{\mathbb{H}}(x,x\cup y)+d_{\mathbb{H}}(y,x\cup y).$

It is straightforward to check that $d_{\mathbb{H}}$ defines a metric on
$\mathbb{H}_{k}$. We can extend $d_{\mathbb{H}}$ on $\mathbb{P}^{1,{\rm
an}}_{k}$ by setting $d_{\mathbb{H}}(x,y)=\infty$ if $x\in\mathbb{P}^{1}(k)$
and $y\in\mathbb{P}^{1,{\rm an}}_{k}$ with $x\neq y$, and
$d_{\mathbb{H}}(x,y)=0$ if $x=y$.

######  Theorem 3.1.

[1, Proposition 2.29] The space $(\mathbb{H}_{k},d_{\mathbb{H}})$ is an
$\mathbb{R}$-tree. The group $\text{PGL}(2,k)$ acts as isometries on
$(\mathbb{H}_{k},d_{\mathbb{H}})$.

Now suppose $k$ is not algebraically closed. Let $\bar{k}^{a}$ be the
completion of an algebraic closure of $k$. Then $\mathbb{P}^{1,{\rm an}}_{k}$
is homeomorphic to the quotient of $\mathbb{P}^{1,{\rm an}}_{\bar{k}^{a}}$ by
the Galois group ${\rm Gal}(\bar{k}^{a}/k)$. Let
$\pi_{\bar{k}^{a}/k}:\mathbb{P}^{1,{\rm
an}}_{\bar{k}^{a}}\to\mathbb{P}^{1,{\rm an}}_{k}$ be the projection map. The
action of the Galois group ${\rm Gal}(\bar{k}^{a}/k)$ on $\mathbb{P}^{1,{\rm
an}}_{\bar{k}^{a}}$ preserves types of points. Therefore, we can define the
type of a point $x\in\mathbb{P}^{1,{\rm an}}_{k}$ by the type of the point
$\pi_{\bar{k}^{a}/k}^{-1}(x)\in\mathbb{P}^{1,{\rm an}}_{\bar{k}^{a}}$. We note
that in general some type I point in $\mathbb{P}^{1,{\rm an}}_{k}$ may not be
rigid.

The action of the Galois group ${\rm Gal}(\bar{k}^{a}/k)$ preserves the
diameter of balls in $\bar{k}^{a}$. Therefore ${\rm Gal}(\bar{k}^{a}/k)$ acts
by isometries on $(\mathbb{H}_{\bar{k}^{a}},d_{\mathbb{H}})$. The set
$\tilde{\mathbb{H}}_{k}\subset\mathbb{H}_{\bar{k}^{a}}$ of fixed points of
this action turns out to be the convex hull $\overline{{\rm
Conv}(\mathbb{P}^{1}(k))}$ of $\mathbb{P}^{1}(k)$ in
$\mathbb{H}_{\bar{k}^{a}}$; see [6, Lemma 1.6]. Hence, in
$\mathbb{P}_{\bar{k}^{a}}^{1,{\rm an}}$, we define

$\widetilde{\mathbb{H}}_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\overline{{\rm
Conv}(\mathbb{P}^{1}(k))}\setminus\mathbb{P}^{1}(k).$

Then $(\widetilde{\mathbb{H}}_{k},d_{\mathbb{H}})$ is a complete metric
$\mathbb{R}$-tree as shown in the previous subsection. Define

$\mathbb{H}_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi_{\bar{k}^{a}/k}(\widetilde{\mathbb{H}}_{k})$

and endow $\mathbb{H}_{k}$ with the metric $d_{\mathbb{H}}$ so that the
projection map
$\pi_{\bar{k}^{a}/k}:(\widetilde{\mathbb{H}}_{k},d_{\mathbb{H}})\to(\mathbb{H}_{k},d_{\mathbb{H}})$
is an isometry. We call $(\mathbb{H}_{k},d_{\mathbb{H}})$ the Berkovic
hyperbolic space of $k$.

The action of $\text{PGL}(2,k)$ on $\mathbb{P}^{1,{\rm an}}_{\bar{k}^{a}}$
preserves types of points; see [1, Proposition 2.15]. Therefore, it also
preserves types of points in $\mathbb{P}^{1,{\rm an}}_{k}$. Morevoer,
$\text{PGL}(2,k)$ acts on $(\mathbb{H}_{k},d_{\mathbb{H}})$ by isometries as
it does on $(\widetilde{\mathbb{H}}_{k},d_{\mathbb{H}})$.

###  3.2. Elements of $\text{PGL}(2,k)$

In this subsection, $k$ is a non-Archimedean local field. Recall that
$\mathcal{O}_{k}$ is the ring of integers of $k$. The first proposition of
this subsection gives a classification of elements in $\text{PGL}(2,k)$.

######  Proposition 3.2 ([6, Proposition 2.2]).

Let $A\in\text{PGL}(2,k)$ and $A\neq{\rm id}$. Then exactly one of the
following holds.

  1. (1)

$|{\rm tr}(\gamma)|>1$: then $A$ is diagonalizable over $k$ and has one
attracting (resp. repelling) fixed point $x_{att}\in\mathbb{P}^{1}(k)$ (resp.
$x_{rep}\in\mathbb{P}^{1}(k)$). Furthermore, for any $x\neq x_{rep}$ in
$\mathbb{P}^{1,{\rm an}}_{k}$, the sequence $A^{n}\cdot x$ converges to
$x_{att}$ as $n\to\infty$. In this case, $A$ is said to be hyperbolic.

  2. (2)

$|{\rm tr}(\gamma)|\leq 1$: then $A$ admits a fixed point in $\mathbb{H}_{k}$.
In this case, $A$ is said to be elliptic. More precisely if

     * •

${\rm tr}^{2}(\gamma)=4$: then $A$ is not diagonalizable, and is conjugate in
$\text{PGL}(2,k)$ to $z\mapsto z+1$; thus it fixes a segment
$[x,y]\subset\mathbb{P}^{1,{\rm an}}_{k}$ where $x$ (resp. $y$) is a type II
(resp. type I) point belonging to the $\text{PGL}(2,k)$ orbit of the Gauss
point. In this case, $A$ is said to be parabolic.

     * •

${\rm tr}^{2}(\gamma)\neq 4$: then in a field extension $K/k$ of degree at
most 2, the matrix $A$ is diagonalizable. In addition, $A$ is conjugate to an
element in $\text{PGL}(2,\mathcal{O}_{k})$ and fixes a type II point in
$\mathbb{H}_{k}$. In this case, $A$ is said to be strictly elliptic.

If $\gamma\in{\rm SL}(2,k)$, we define the norm $||\gamma||$ of $\gamma$ to be
the largest absolute value of entries in $\gamma$. The next proposition gives
the Cartan KAK decomposition of elements in $\text{PGL}(2,k)$.

######  Proposition 3.3 (Cartan decomposition, [6, Proposition 2.4]).

Let $\gamma\in{\rm SL}(2,k)$. Then there exist $m,n\in{\rm
SL}(2,\mathcal{O}_{k})$ and $a={\rm diag}(\lambda,\lambda^{-1})$ where
$\lambda\in k$ and $|\lambda|\geq 1$ such that $\gamma=man$. Furthermore
$||\gamma||=||a||=|\lambda|$.

###  3.3. Projections and cross ratios

In this subsection, $(k,|\cdot|)$ is a non-Archimedean local field with the
absolute value compatible with the Haar measure as in Lemma 2.2. Let
$\alpha\in\mathbb{P}^{1}(k)$ and consider the geodesic
$[0,\infty]\subset\mathbb{P}^{1,{\rm an}}_{k}$ from $0\in\mathbb{P}^{1}(k)$
and $\infty\in\mathbb{P}^{1}(k)$. We say that $\beta\in[0,\infty]$ is the
projection of $\alpha$ onto $[0,\infty]$ if $\beta$ is the unique point in
$[0,\infty]$ such that there exists a geodesic segment
$[\beta,\alpha]\subset\mathbb{P}^{1,{\rm an}}_{k}$. The following lemma gives
a formula for $\beta$ in terms of balls in $\mathbb{P}^{1}(k)$.

######  Lemma 3.4.

Let $\alpha\in\mathbb{P}^{1}(k)$. Then the projection of $\alpha$ onto the
geodesic $[0,\infty]\subset\mathbb{P}^{1,{\rm an}}_{k}$ is the ball
$B(0,|\alpha|)$.

###### Proof.

The point $\alpha\in\mathbb{P}^{1}(k)$ corresponds to the ball $B(\alpha,0)$
in $k$, which is disjoint from the ball $B(0,0)$. Therefore the point of
projection of $\alpha$ onto $[0,\infty]$ is $B(0,|\alpha-0|)$. ∎

######  Lemma 3.5.

If $a,b\in\mathbb{P}^{1}(k)$ satisfy $0<|a|<|b|$, then we have

$d_{\mathbb{H}}(B(0,|a|),B(0,|b|))=\log_{v}\left|[\infty,0;|a|,|b|]\right|$

where $[u,v;x,y]$ is the cross ratio on $\mathbb{P}^{1}(k)$.

###### Proof.

If $0<|a|<|b|$, the geodesic segment
$[B(0,|a|),B(0,|b|)]\subset[0,\infty]\subset\mathbb{P}^{1,{\rm an}}_{k}$ is
isometric to the interval $[\log_{v}|a|,\log_{v}|b|]\subset\mathbb{R}$
equipped with the Lebesgue measure. Therefore, we have

$d_{\mathbb{H}}(B(0,|a|),B(0,|b|))=\log_{v}|b|-\log_{v}|a|=\log_{v}\left|[\infty,0;|a|,|b|]\right|.$

∎

Let $P:\mathbb{P}^{1}(k)\to[0,\infty]$ be the projection map sending
$\alpha\in\mathbb{P}^{1}(k)$ to its projection $B(0,|\alpha|)$ on
$[0,\infty]$.

######  Lemma 3.6.

The projection map $P:\mathbb{P}^{1}(k)\to[0,\infty]$ maps a set of Haar
measure zero to a set of Lebesgue measure zero on $[0,\infty]$.

###### Proof.

Let $E\subset\mathbb{P}^{1}(k)$ be a set of Haar measure zero. We first note
that

$|P(x)-P(y)|\leq|x-y|$

for any $x,y\in E$. Indeed, $P(z)=|z|$ for any $z\in\mathbb{P}^{1}(k)$ and the
conclusion follows from the triangle inequality.

Fix $\epsilon>0$ and let $m$ denote the Haar measure. Since $m(E)=0$, we can
cover $E$ by balls $B(x_{i},r_{i})$ with $x_{i}\in E$ and
$\sum_{i}m(B(x_{i},r_{i}))<\epsilon.$ In particular, we can choose
$r_{i}\in|k^{\times}|$ with

$\sum_{i}r_{i}<\frac{\epsilon}{m_{b}}$

where $m_{b}>0$ is the Haar measure of a ball of radius 1, since Lemma 2.2
gives $m(B(x_{i},r_{i}))=r_{i}m(B(x_{i},1))$ and therefore we have

$\sum_{i}m(B(x_{i},r_{i}))=m_{b}\sum_{i}r_{i}<\epsilon.$

If $x\in E\cap B(x_{i},r_{i})$, then $|x_{i}-x|<r_{i}$. It follows that
$|P(x_{i})-P(x)|\leq|x_{i}-x|<r_{i}$ and consequently that $P(E\cap
B(x_{i},r_{i}))\subset B(P(x_{i}),r_{i})$. Hence,

$P(E)=\cup_{i}P(E\cap B(x_{i},r_{i}))\subset\cup_{i}B(P(x_{i}),r_{i}).$

Let $\mu$ be the Lebesgue measure on $\mathbb{R}$. We have

$\displaystyle\mu(P(E))$ $\displaystyle\leq\sum_{i}\mu(B(P(x_{i}),r_{i}))$
$\displaystyle=2\sum_{i}r_{i}<\frac{2\epsilon}{m_{b}}.$

Therefore $\mu(P(E))=0$. ∎

##  4\. Non-Archimedean Basmajian’s identity

In this section, we prove Theorem 1.1. The proof adapts the idea of
Basmajian’s decomposition; namely, we decompose an interval in the real line
into a set of measure zero which corresponds to the limit set, plus a
countable collection of disjoint intervals, one for each double coset, whose
length is given by log of the corresponding cross ratio.

Let $\Sigma$ be a connected compact oriented surface with boundary
$\partial\Sigma$ whose double $\hat{\Sigma}$ has genus at least 2. Let
$\alpha_{1},\ldots,\alpha_{k}\in\pi_{1}\Sigma$ represent the free homotopy
classes of boundaries $a_{1},\ldots,a_{k}$. Each boundary $a_{j}$ is oriented
so that the surface is on the left. Fix a finite area hyperbolic structure on
$\Sigma$ so that $\partial\Sigma$ is totally geodesic. Let
$V\subset\mathbb{H}^{2}$ be the universal cover of $\Sigma$ so that
$\partial\pi_{1}\Sigma=\partial_{\infty}V=\overline{V}\cap\mathbb{S}^{1}$. Fix
a boundary component $\alpha_{j}$ of $\Sigma$ and let
$\alpha_{j}^{+},\alpha_{j}^{-}\in\partial\pi_{1}\Sigma$ be the attracting and
repelling fixed points of $\alpha_{j}$, respectively.

Let $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ be projective Anosov. By
Proposition 2.7, $\rho$ is the restriction to $\pi_{1}\Sigma$ of some
projective Anosov representation
$\hat{\rho}:\pi_{1}\hat{\Sigma}\to\text{PSL}(2,k)$. Therefore we can consider
the interval $(\alpha_{j}^{-},\alpha_{j}^{+})$ as a subset of $\mathbb{S}^{1}$
with the given orientation. In the following, we denote by
$\zeta^{1}=\zeta^{1}_{\hat{\rho}}$. We conjugate $\hat{\rho}$ so that
$\zeta^{1}(\alpha_{j}^{-})=0$ and $\zeta^{1}(\alpha_{j}^{+})=\infty$.
Therefore $[0,\infty]\subset\mathbb{H}_{k}$ is the axis of $\alpha_{j}$.

Recall that $P:\mathbb{P}^{1}(k)\to[0,\infty]$ is the projection onto the axis
$[0,\infty]$.

######  Lemma 4.1.

Let $x\in(\alpha_{j}^{-},\alpha_{j}^{+})\subset\mathbb{S}^{1}$. Then we have

$\ell_{\rho}(\alpha_{j})=d_{\mathbb{H}}(P\circ\zeta^{1}(\alpha_{j}\cdot
x),P\circ\zeta^{1}(x)).$

###### Proof.

We compute

$\displaystyle d_{\mathbb{H}}(P\circ\zeta^{1}(\alpha_{j}\cdot
x),P\circ\zeta^{1}(x))$ $\displaystyle=\log_{v}|\zeta^{1}(\alpha_{j}\cdot
x)|-\log_{v}|\zeta^{1}(x)|$
$\displaystyle=\log_{v}\left|C_{\rho}(\alpha_{j}^{+},\alpha_{j}^{-};x,\alpha_{j}\cdot
x)\right|$ $\displaystyle=\ell_{\rho}(\alpha_{j}).$

∎

Therefore
$\mathbb{T}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(0,\infty)/\ell_{\rho}(\alpha_{j})\mathbb{Z}$
is a fundamental domain of $\alpha_{j}$ in $\mathbb{H}_{k}$. Let
$\pi:(0,\infty)\to\mathbb{T}$ be the projection map.

######  Lemma 4.2.

We have the following decomposition of $\mathbb{T}$:

$\mathbb{T}=\pi(P\circ\zeta^{1}(\partial_{\infty}V))\sqcup\left(\bigsqcup_{q=1}^{k}\bigsqcup_{w\in\mathcal{L}_{j,q}}I_{w}\right)$

where $I_{w}\subset\mathbb{T}$ is an interval and $\mathcal{L}_{j,q}$ is the
set of double cosets
$\langle\alpha_{j}\rangle\backslash\pi_{1}\Sigma/\langle\alpha_{q}\rangle$.

###### Proof.

We first observe that $\mathbb{S}^{1}\setminus\partial_{\infty}V$ is a
disjoint union of intervals of the form
$\tilde{I}_{\beta}=(\beta^{-},\beta^{+})$ where $\beta=g\alpha_{k}g^{-1}$ for
some boundary element $\alpha_{k}$ and $g\in\pi_{1}\Sigma$. Indeed, the
endpoints $\beta^{-}$ and $\beta^{+}$ are the endpoints of lifts of boundary
elements. We claim that each interval $\tilde{I}_{\beta}$ correspond to a
coset $gH_{k}$ in $\bigsqcup_{1\leq q\leq m}\pi_{1}\Sigma/H_{q}.$ Recall that
$H_{k}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\langle\alpha_{k}\rangle$.
Indeed, if
$\tilde{I}_{g_{1}\alpha_{k}g_{1}^{-1}}=\tilde{I}_{g_{2}\alpha_{l}g_{2}^{-1}}$,
then we know that

$g_{1}\alpha_{k}^{\pm}=g_{2}\alpha_{l}^{\pm}$

and therefore $(g_{2}^{-1}g_{1})\alpha_{k}^{\pm}=\alpha_{l}^{\pm}$. Since
$\alpha_{k}$ and $\alpha_{l}$ are boundary elements, it follows that $k=l$ and
therefore $g_{2}^{-1}g_{1}\in H_{k}$.

Let
$\hat{I}_{\beta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(P\circ\zeta^{1}(\beta^{-}),P\circ\zeta^{1}(\beta^{+}))\subset[0,\infty]$.
We claim that $\hat{I}_{\beta}\cap\hat{I}_{\beta^{\prime}}=\emptyset$ for
$\beta\neq\beta^{\prime}$. Indeed, otherwise the axis of $\beta$ in
$\mathbb{H}_{k}$ and the axis of $\beta^{\prime}$ will intersect. But this is
impossible as $\beta$ and $\beta^{\prime}$ represent boundary elements which
are disjoint.

Let
$I_{\beta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\pi(\hat{I}_{\beta})$.
Then we have

$\mathbb{T}=\pi(P\circ\zeta^{1}(\partial_{\infty}V))\sqcup\left(\bigsqcup_{\beta}I_{\beta}\right)$

where each $\beta$ corresponds to a double coset in $\bigsqcup_{1\leq q\leq
k,q\neq j}H_{j}\backslash\pi_{1}\Sigma/H_{q}.$ ∎

Let $\nu_{\mathbb{H}}$ be the hyperbolic measure on $[0,\infty]$; namely,
$\nu_{\mathbb{H}}((a,b))=\log_{v}(b/a)$ for any interval $(a,b)$ with
$0<a<b<\infty$.

######  Lemma 4.3.

We have $\nu_{\mathbb{H}}(\mathbb{T})=\ell_{\rho}(\alpha_{j})$ and

$\nu_{\mathbb{H}}(I_{\beta})=\log_{v}\left|C_{\rho}(\alpha_{j}^{+},\alpha_{j}^{-};w\cdot\alpha_{j}^{+},w\cdot\alpha_{j}^{-})\right|.$

###### Proof.

By construction of $\mathbb{T}$, we have
$\nu_{\mathbb{H}}(\mathbb{T})=\ell_{\rho}(\alpha_{j})$. We compute

$\displaystyle\nu_{\mathbb{H}}(I_{\beta})$
$\displaystyle=\log_{v}|\zeta^{1}(\beta^{-})|-\log_{v}|\zeta^{1}(\beta^{+})|$
$\displaystyle=\log_{v}|[\infty,0;\zeta^{1}(\beta^{+}),\zeta^{1}(\beta^{-})]|$
$\displaystyle=\log_{v}\left|C_{\rho}(\alpha_{j}^{+},\alpha_{j}^{-};w\cdot\alpha_{j}^{+},w\cdot\alpha_{j}^{-})\right|$

where $w$ is the unique representative of the double coset corresponding to
$I_{\beta}$. Here $w$ is unique if we put an ordering on a symmetric
generating set of $\pi_{1}\Sigma$ and let $w$ be the double coset
representative that is the shortest in word length and the largest with
respect to the ordering. ∎

In the next two lemmas, we show that the set
$\pi(F_{\rho}(\partial_{\infty}V))$ has $\nu_{\mathbb{H}}$ measure zero.

######  Lemma 4.4.

Let $\rho:\pi_{1}\Sigma\to\text{PSL}(2,k)$ be projective Anosov and let $m$ be
the Haar measure on $k$. Then we have

$m(\zeta_{\rho}^{1}(L))=0$

where $L\subset\partial\pi_{1}\Sigma$ is the limit set of $\pi_{1}\Sigma$.

###### Proof.

Let $L\subset\partial\pi_{1}\Sigma$ be the limit set of $\pi_{1}\Sigma$. Since
$\pi_{1}\Sigma$ is a free group, we have

$L=\bigsqcup_{g_{i}}g_{i}\cdot L$

where $g_{i}$ runs through the generators of $\pi_{1}\Sigma$ and their
inverses.

Then the limit set
$\Lambda\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\zeta_{\rho}^{1}(L)\subset\mathbb{P}^{1}(k)$
is the union of

$\Lambda=\bigsqcup_{g_{i}}\zeta_{\rho}^{1}(g_{i}\cdot
L)=\bigsqcup_{g_{i}}\rho(g_{i})\cdot\Lambda$

since $\zeta^{1}$ is equivariant. Let $m$ be the Haar measure on $k$. Then

$m(\Lambda)=\sum_{g_{i}}m(\rho(g_{i})\cdot\Lambda).$

We conjugate the representation so that $\rho(g_{1})$ is the diagonal matrix
$\begin{bmatrix}\lambda&0\\\ 0&\lambda^{-1}\end{bmatrix}$ with $|\lambda|>1$.
Then by Lemma 2.2 we have

$m(\rho(g_{1})\cdot\Lambda)=m(\lambda^{2}\cdot\Lambda)=|\lambda^{2}|m(\Lambda)$

and therefore

$0=(|\lambda|^{2}-1)m(\Lambda)+\sum_{g_{i}\neq
g_{1}}m(\rho(g_{i})\cdot\Lambda).$

Since each term in the summand is non-negative and they sum up to 0, we know
that in fact each term is equal to 0. In particular, $m(\Lambda)=0$. ∎

######  Lemma 4.5.

We have $\nu_{\mathbb{H}}(\pi(P\circ\zeta^{1}(\partial_{\infty}V)))=0$.

###### Proof.

Note that the set $P\circ\zeta^{1}(\partial_{\infty}V)\subset(0,\infty)$ has
Lebesgue measure zero by Lemmas 4.4 and 3.6. Since the projection map
$\pi:(0,\infty)\to\mathbb{T}$ is locally an isometry, the set
$\pi(P\circ\zeta^{1}(\partial_{\infty}V))$ has Lebesgue measure zero. Finally,
the hyperbolic measure of a set of Lebesgue measure zero is also zero. ∎

Now we give a proof of Theorem 1.1.

###### Proof of Theorem 1.1.

For boundary $\alpha_{j}$, by Lemmas 4.2, 4.3 and 4.5, we have

$\displaystyle\ell_{\rho}(\alpha_{j})=\nu_{\mathbb{H}}(\mathbb{T})$
$\displaystyle=\nu_{\mathbb{H}}(\pi(F_{\rho}(\partial_{\infty}V)))+\sum_{\beta}\nu_{\mathbb{H}}(I_{\beta})$
$\displaystyle=0+\sum_{q=1}^{k}\sum_{w\in\mathcal{L}_{j,q}}\log_{v}\left|C_{\rho}(\alpha_{j}^{+},\alpha_{j}^{-};w\cdot\alpha_{j}^{+},w\cdot\alpha_{j}^{-})\right|$

The identity is obtained by summing over $\alpha_{j}$’s. ∎

## References

  * [1] M. Baker and R. Rumely, Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, vol. 159, American Mathematical Society, 2010.
  * [2] A. Basmajian, The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math., 115 (1993), 1139–1159.
  * [3] J. Bochi, R. Potrie and A. Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc. (JEMS)., 21 (2019), 3343–3414.
  * [4] M. Bridgeman and S. P. Tan, Identities on hyperbolic manifolds, The Handbook of Teichmüller Theory, Vol. 5 (2016), 19-53.
  * [5] J. W. S. Cassels, Local fields, London Mathematical Society Student Texts 3, Cambridge University Press, Cambridge, 1986.
  * [6] R. Dujardin and C. Favre, Degenerations of ${\rm SL}(2,\mathbb{C})$ representations and Lyapunov exponents, Annales Henri Lebesgue, 2 (2019), 515-565.
  * [7] E. Falbel, A. Guilloux and P. Will, Hilbert metric, beyond convexity, arXiv:1804.05317.
  * [8] F. Fanoni and M. B. Pozzetti, Basmajian-type inequalities for maximal representations, J. Differential Geom., 116 (2020), 405-458.
  * [9] Y. M. He, Basmajian-type identities and Hausdorff dimension of limit sets, Ergodic Theory Dynam. Systems, 38 (2018), 2224-2244.
  * [10] Y. M. He, Identities for hyperconvex Anosov representations, Journal of Topology and Analysis. https://doi.org/10.1142/S1793525322500054
  * [11] Y. Huang, R. C. Penner and A. M. Zeitlin, Super McShane identity, arXiv:1907.09978.
  * [12] Y. Huang and Z. Sun, McShane identities for higher Teichmüller theory and the Goncharov-Shen potential, to appear Mem. Amer. Math. Soc.
  * [13] F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math., 165 (2006), 51-114.
  * [14] F. Labourie, Cross ratios, surface groups, PSL$(n,\mathbb{R})$ and diffeomorphisms of the circle, Publ. Math. Inst. Hautes Etudes Sci., 106 (2007), 139-213.
  * [15] F. Labourie and G. McShane, Cross ratios and identities for higher Teichmüller-Thurston theory, Duke Math. J., 149 (2009), 279-345.
  * [16] M. B. Pozzetti, A. Sambarino and A. Wienhard, Conformality for a robust class of non-conformal attractors, J. Reine Angew. Math. 774 (2021), 1-51.
  * [17] J-P. Serre, Local fields, Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979.
  * [18] N. Vlamis and A. Yarmola, Basmajian’s identity in higher Teichmüller-Thurston theory, J. Topol., 10 (2017), 744-764.