# Positive crossratios, barycenters, trees and applications to maximal
representations

M. Burger  Department Mathematik, ETH Zentrum, Rämistrasse 101, CH-8092
Zürich, Switzerland<EMAIL_ADDRESS>,  A. Iozzi  Department Mathematik,
ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland<EMAIL_ADDRESS>,  A. Parreau  Institut Fourier, CS 40700, 38058 Grenoble cedex 09, France
<EMAIL_ADDRESS>and  M. B. Pozzetti  Mathematical
Institute, Heidelberg University, Im Neuenheimerfeld 205, 69120 Heidelberg,
Germany<EMAIL_ADDRESS>
###### Abstract.

We study metric properties of maximal framed representations of fundamental
groups of surfaces in symplectic groups over real closed fields, interpreted
as actions on Bruhat–Tits buildings endowed with adapted Finsler norms. We
prove that the translation length can be computed as intersection with a
geodesic current, give sufficient conditions guaranteeing that such a current
is a multicurve, and, if the current is a measured lamination, construct an
isometric embedding of the associated tree in the building. These results are
obtained as application of more general results of independent interest on
positive crossratios and actions with compatible barycenters.

Marc Burger and Alessandra Iozzi were partially supported by the SNF grant
2-77196-16. Alessandra Iozzi acknowledges moreover support from U.S. National
Science Foundation grants DMS 1107452, 1107263, 1107367 ”RNMS: Geometric
Structures and Representation Varieties” (the GEAR Network). Beatrice Pozzetti
is supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence
Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of
Excellence), and acknowledges further support by DFG grant 338644254 (within
the framework of SPP2026). Marc Burger, Alessandra Iozzi and Beatrice Pozzetti
would like to thank the Isaac Newton Institute for Mathematical Sciences,
Cambridge, for support and hospitality during the programme “Non-Positive
Curvature Group Actions and Cohomology” where work on this paper was
undertaken. This work was supported by EPSRC grant no P/K032208/1.

##  1\. Introduction

### Maximal framed representations in real closed fields

Let $\Sigma:=\Gamma\backslash{\mathcal{H}}^{2}$ be the quotient of the
Poincaré upper half plane ${\mathcal{H}}^{2}$ by a torsion-free lattice
$\Gamma<\operatorname{PSL}(2,\mathbf{R})$ and let $G$ be a simple real
algebraic group. The aim of Higher Teichmüller Theory is to single out and
study special components or specific semialgebraic subsets of the
representation variety $\mathrm{Hom}(\Gamma,G)$ that consist of injective
homomorphisms with discrete image; such components thus generalize the
Teichmüller space. Prominent examples are Hitchin components for real split
groups $G$ (for example $G=\operatorname{PSL}(n,\mathbf{R})$ or
$\operatorname{PSp}(2n,\mathbf{R})$), maximal representations for Hermitian
groups $G$ (for example $G=\operatorname{PU}(p,q)$ or
$\operatorname{PSp}(2n,\mathbf{R})$) and $\Theta$-positive representations for
$G=\operatorname{PO}(p,q)$. The goal of this paper is to study asymptotic
properties of maximal representations into $\operatorname{PSp}(2n,\mathbf{R})$
with the aid of geodesic currents. This applies in particular to the
$\operatorname{PSp}(2n,\mathbf{R})$-Hitchin component if $\Sigma$ is compact.

In the study of appropriate compactifications of character varieties,
representations of $\Gamma$ into algebraic groups over non-Archimedean real
closed fields play an important role [Bru88, Ale08, Par12, BP17, BIPP21c]: for
example, ultralimits of representations in $\operatorname{PSp}(2n,\mathbf{R})$
can be understood as representations
$\rho_{\omega,\sigma}\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{R}_{\omega,\sigma})$,
where $\mathbf{R}_{\omega,\sigma}$ is a Robinson field (see §8.1 and [Par12]).
However, the viewpoint of the real spectrum compactification of character
varieties leads typically to the study of representations into real closed
fields $\mathbf{F}$ that are small when compared to Robinson fields, for
example $\mathbf{F}$ is often of finite transcendence degree over the field of
real algebraic numbers.

Given a representation $\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$
where $\mathbf{F}$ is a general real closed field, having a maximal Toledo
invariant as defined in [BIPP21c, Definition 18], admitting a maximal framing
defined on a $\Gamma$-invariant non-empty subset of
$\partial{\mathcal{H}}^{2}$, or admitting a maximal framing defined on the set
of fixed points of hyperbolic elements are all equivalent conditions [BIPP21b]
(see [BIPP21c, Theorem 20] for a precise statement). This paper solely relies
upon the third definition, which we now recall. If $\mathbf{F}^{2n}$ is
endowed with the standard symplectic form, let $\mathcal{L}(\mathbf{F}^{2n})$
be the space of Lagrangians in $\mathbf{F}^{2n}$. The Maslov cocycle (see §6.4
and [LV80]) classifies the orbits of $\operatorname{PSp}(2n,\mathbf{F})$ on
triples of pairwise transverse Lagrangians. Such a triple is maximal if the
cocycle takes its maximal value $n$. Let $\partial{\mathcal{H}}^{2}$ be the
boundary of the hyperbolic plane, which we endow with the cyclic ordering on
triples of points induced by the orientation of ${\mathcal{H}}^{2}$.

######  Definition 1.1.

Let ${H_{\Gamma}}\subset\partial{\mathcal{H}}^{2}$ be the set of fixed points
of hyperbolic elements of $\Gamma$. A representation
$\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$ is _maximal framed_ if
there is a $\rho$-equivariant map
$\varphi\colon{H_{\Gamma}}\to\mathcal{L}(\mathbf{F}^{2n})$ sending positively
oriented triples in ${H_{\Gamma}}$ to maximal triples of Lagrangians.

We assume that the real closed field $\mathbf{F}$ admits an order compatible
$\mathbf{R}$-valued valuation $v$ with value group
$\Lambda=v(\mathbf{F}^{\times})<\mathbf{R}$. Then the group
$\operatorname{PSp}(2n,\mathbf{F})$ acts by isometries on a $\Lambda$-metric
space111In [BIPP21c] denoted with
$\mathcal{B}_{\operatorname{PSp}(2n,\mathbf{F})}$.
$\mathcal{B}_{n}^{\mathbf{F}}$ obtained as quotient of the Siegel upper half
space $\mathcal{S}^{n}_{\mathbf{F}}$ associated to
$\operatorname{PSp}(2n,\mathbf{F})$ (see § 6.2 and [BIPP21c, § 3.4]). If
$\mathbf{F}=\mathbf{R}$, $\mathcal{B}_{n}^{\mathbf{R}}$ coincides with
$\mathcal{S}^{n}_{\mathbf{R}}$, while if $\mathbf{F}$ is non-Archimedean this
metric space $\mathcal{B}_{n}^{\mathbf{F}}$ sits naturally inside the Bruhat-
Tits building associated to $\operatorname{PSp}(2n,\mathbf{F})$ as a dense
subset.222Note, however, that the metric that we consider here is only
biLipschitz to the restriction of the CAT(0) metric on the Bruhat-Tits
building. See §6.2 for the Finsler metric relevant to our purposes. The latter
relationship will play no role in this paper, but will be discussed in detail
in [BIPP21a] (see also [BIPP21c, §2.1] and [KT04]). The translation length of
an element $g\in\operatorname{PSp}(2n,\mathbf{F})$ acting on
$\mathcal{B}_{n}^{\mathbf{F}}$ induces then the _length function_

$L(g):=-2\sum_{i=1}^{n}v(|\lambda_{i}|)\,,$

where
$\lambda_{1},\dots,\lambda_{n},\lambda_{n}^{-1},\dots,\lambda_{1}^{-1}\in\mathbf{F}(\sqrt{-1})$
are the eigenvalues of a representative
$\overline{g}\in\operatorname{Sp}(2n,\mathbf{F})$ of
$g\in\operatorname{PSp}(2n,\mathbf{F})$ counted with multiplicity and ordered
in such a way that $|\lambda_{1}|\geq\dots\geq|\lambda_{n}|\geq 1$. Here we
denote by $|\cdot|\colon\mathbf{F}(\sqrt{-1})\to\mathbf{F}_{+}$ the absolute
value, that is the square root of the norm function on the quadratic extension
$\mathbf{F}(\sqrt{-1})$ of $\mathbf{F}$.

In our first result we construct a geodesic current on $\Sigma$ encoding the
length function $\gamma\mapsto L(\rho(\gamma))$ of a maximal framed
representation. Recall that a _geodesic current_ is a $\Gamma$-invariant
positive Radon measure on the space $(\partial{\mathcal{H}}^{2})^{(2)}$ of
pairs of distinct points in $\partial{\mathcal{H}}^{2}$. The _Bonahon
intersection_ $i(\mu,\lambda)$ of two geodesic currents $\mu$ and $\lambda$
extends the topological intersection number of homotopy classes of curves on
$\Sigma$: in fact a (non-oriented) closed geodesic $c\subset\Sigma$ gives rise
to the current $\delta_{c}:=\frac{1}{2}\sum_{(a,b)}\delta_{(a,b)}$, where we
sum on the set of oriented geodesics
$(a,b)\in(\partial{\mathcal{H}}^{2})^{(2)}$ lifting $c$. For such currents
$i(\delta_{c},\delta_{c^{\prime}})$ is the topological intersection number of
$[c]$ and $[c^{\prime}]$.

######  Theorem 1.2.

Let $\mathbf{F}$ be a real closed field with an order compatible valuation $v$
and let $\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$ be maximal
framed. Then there is a geodesic current $\mu_{\rho}$ such that, for any
closed geodesic $c\subset\Sigma$ and for every $\gamma\in\Gamma$ representing
$c$,

(1) $i(\mu_{\rho},\delta_{c})=L(\rho(\gamma))\,.$

The current $\mu_{\rho}$ is non-zero if and only if there exists
$\gamma\in\Gamma$ with $v(\mathrm{tr}(\rho(\gamma)))<0$.

If $n=1$, $\mathbf{F}=\mathbf{R}$ and
$\rho\colon\Gamma\to\operatorname{PSL}(2,\mathbf{R})$ is the lattice
embedding, then $\mu_{\rho}$ is the Liouville current [Bon88], that is the
unique $\operatorname{PSL}(2,\mathbf{R})$-invariant geodesic current. If
$\Sigma$ is compact and $\mathbf{F}=\mathbf{R}$, Theorem 1.2 was proven by
Martone–Zhang [MZ19, Theorem 1.1]. Notice that in the case of
$\operatorname{SL}(2,\mathbb{F})$ $\mu_{\rho}$ is always a measured lamination
(see § 8.2).

For the next result we will need the notion of systole of a maximal framed
representation $\rho$ (following [BIPP19])

$\mathrm{Syst}(\rho):=\inf\;\\{L(\rho(\gamma)):\,\gamma\in\Gamma,\,\gamma\text{
hyperbolic}\\}\,.$

The systole of any real maximal framed representation
$\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{R})$ is positive (see §
7.3). On the other hand, for non-Archimedean real closed fields $\mathbf{F}$,
many different possibilities can happen: if $\Sigma$ is compact, all maximal
framed representations $\rho\colon\Gamma\to\operatorname{SL}(2,\mathbf{F})$
have vanishing systole (since $\mu_{\rho}$ is a measured lamination), while,
in higher rank, there are many examples of non-Archimedean maximal framed
representations with positive systole (see [BIPP19, Corollary 1.11]). For
these representations we have:

######  Corollary 1.3.

Assume that $\Sigma$ is compact, and $\mathbf{F}$ is non-Archimedean. Let
$\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$ be a maximal framed
representation. If $\mathrm{Syst}_{\Sigma}(\rho)>0$, then for every
$x\in\mathcal{B}_{n}^{\mathbf{F}}$ the orbit map

$\displaystyle\Gamma$
$\displaystyle\longrightarrow\,\,\mathcal{B}_{n}^{\mathbf{F}}$ $\displaystyle
x$ $\displaystyle\longmapsto\rho(\gamma)x$

is a quasi-isometric embedding.

We now give a robust criterion guaranteeing that the current $\mu_{\rho}$ is
atomic. We say that a geodesic current is a _multicurve_ if it is a finite sum
of $\Gamma$-orbits of Dirac masses on (lifts of) closed geodesics and
geodesics with endpoints in cusps.

######  Theorem 1.4.

Let $\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$ be maximal framed
and let $\mathbf{Q}(\rho)<\mathbf{F}$ be the field generated over $\mathbf{Q}$
by matrix coefficients of $\rho$. If the restriction of
$v\colon\mathbf{F}^{\times}\to\mathbf{R}$ to $\mathbf{Q}(\rho)$ is discrete,
then, up to rescaling, the associated current $\mu_{\rho}$ is a multicurve.

Using Strubel coordinates we construct examples of representations to which
Theorem 1.4 applies (see §8.3). Moreover we prove in [BIPP21b] that a maximal
framed representation $\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$ is
always conjugate to a representation
$\rho^{\prime}\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F}_{1})$ for a
finite extension $\mathbf{F}_{1}$ of the field $\mathbf{Q}(\mathrm{tr}(\rho))$
generated by the traces of the representation $\rho$. As a result, Theorem 1.4
applies as soon as the field $\mathbf{Q}(\mathrm{tr}(\rho))$ generated by the
traces of the representation $\rho$ has discrete valuation; using this we show
in [BIPP21b] that multicurves are dense in both the real spectrum and Weyl
chamber length compactifications of character varieties of maximal
representations.

We turn now to the case in which the current $\mu_{\rho}$ in Theorem 1.2 is a
measured lamination. In this case we denote by $\mathcal{T}(\mu_{\rho})$ the
associated $\mathbf{R}$-tree, and by $\mathcal{V}(\mu_{\rho})$ its vertex set
(see § 2.3 for the definition). We have then

######  Theorem 1.5.

Let $\mathbf{F}$ be non-Archimedean real closed and let
$\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$ be a maximal framed
representation. If the associated current $\mu_{\rho}$ is a measured
lamination, then there is a $\Gamma$-equivariant isometric embedding

$\mathcal{V}(\mu_{\rho})\hookrightarrow{\mathcal{B}_{n}^{\mathbf{F}}}.$

We will see that if $\mathbf{Q}(\rho)$ has discrete valuation, then
$\mathcal{V}(\mu_{\rho})$ is the vertex set of a simplicial tree (see § 7.3
for a general statement).

### Currents associated to positive crossratios

The proof of Theorem 1.2 relies on an abstract framework that is applicable to
more general situations and that we shortly describe here. Let
$X\subset\partial{\mathcal{H}}^{2}$ be a $\Gamma$-invariant non-empty subset,
such as, for example the set ${H_{\Gamma}}$ of fixed points of hyperbolic
elements in $\Gamma$ and let $X^{[4]}$ denote the set of positively ordered
quadruples in $X$. A positive crossratio is a $\Gamma$-invariant function

$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]\colon
X^{[4]}\longrightarrow[0,\infty),$

that is flip-invariant

$[x_{1},x_{2},x_{3},x_{4}]=[x_{3},x_{4},x_{1},x_{2}]$

and satisfies the property

$[x_{1},x_{2},x_{4},x_{5}]=[x_{1},x_{2},x_{3},x_{5}]+[x_{1},x_{3},x_{4},x_{5}]\,$

whenever defined. Our definition is considerably more general than others
existing in the literature (see Remark 3.2 for a comparison). First we only
require our crossratio to be defined on a dense subset $X$ of the boundary of
the hyperbolic plane: this is important for some of the applications. For
example, if the field $\mathbf{F}$ is countable, also
$\mathcal{L}(\mathbf{F}^{n})$ is countable, and thus a maximal framing
associated to a representation
$\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$, as well as the induced
crossratio, can only be defined on a countable set. Furthermore some
representations, as for example those defined via Fock–Goncharov or shear
coordinates, only have a framing defined on a countable set not including the
fixed points of hyperbolic elements. Second we do not require any continuity
on our crossratio. In many interesting examples, the crossratios arising from
representations over non-Archimedean real closed fields are integer valued,
and thus cannot be continuous. Dropping the continuity assumption on the
crossratio also allows us to us encompass the theory of crossratios arising
from actions on trees (see Example 3.5).

If $\gamma\in\Gamma$ is hyperbolic and $\\{\gamma_{-},\gamma_{+}\\}\subset X$,
the period333See § 3.3 for a more general definition of the period without the
restriction that $\\{\gamma_{-},\gamma_{+}\\}\subset X$.
$\mathrm{per}(\gamma)$ of $\gamma$ with respect to
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ is defined by

$\mathrm{per}(\gamma)\colon=[\gamma_{-},x,\gamma x,\gamma_{+}]\,,$

where $x\in X$ is any point such that $(\gamma_{-},x,\gamma x,\gamma_{+})\in
X^{[4]}$. We show:

######  Theorem 1.6.

Let $X\subset\partial{\mathcal{H}}^{2}$ be a $\Gamma$-invariant non-empty
subset and $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ a positive crossratio
on $X$. Then there is a geodesic current $\mu$ on $\Sigma$ such that for all
hyperbolic $\gamma\in\Gamma$

$\mathrm{per}(\gamma)=i(\mu,\delta_{c}).$

The geodesic current $\mu$ depends continuously on the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$.

The theorem has been previously shown by Martone–Zhang under the hypothesis
that $X=\partial{\mathcal{H}}^{2}$ and the crossratio is continuous [MZ19].
For the last statement we consider the space $\mathcal{CR}^{+}(X)$ of positive
crossratios as a closed convex cone in the topological vector space of
crossratios on $X$ with the topology of pointwise convergence. This last
property will be used in the proof of the continuity of the map which to a
point in the real spectrum compactification of maximal representations
associates a geodesic current [BIPP21a]; see also [BIPP21c, Theorem 36].

The proof of Theorem 1.6 bypasses the possible discontinuities of the
crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ by forcing inner and
outer regularity of the current $\mu$ and using its $\sigma$-additivity. As an
application of the explicit construction we obtain:

######  Corollary 1.7.

If the crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ is integral
valued, then the current $\mu$ is a multicurve.

To deduce Theorems 1.2 and 1.4, from Theorem 1.6 and Corollary 1.7, we use the
maximal framing to construct a positive crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\rho}$ on ${H_{\Gamma}}$ whose
periods satisfy the equality $\mathrm{per}(\gamma)=L(\rho(\gamma))$. Then
Theorem 1.6 provides a geodesic current with the required properties.

Maximal representations are not the only class of representations whose length
function is given by the periods of a positive crossratio: this is the case
for all _positively ratioed representations_ [MZ19] – a class that also
includes Hitchin representations [Lab07] –, representations satisfying
property $H_{k}$ [BP20] and $\Theta$-positive representations [BP21].
Corollary 1.7 can be used to study asymptotic properties of these
representations as well.

Our approach using $\sigma$-additivity of geodesic currents has interesting
applications even for representations in $\operatorname{PSp}(2n,\mathbf{R})$,
for which we cannot always assume that the crossratio is continuous. The
simplest instance is for $\operatorname{PSL}(2,\mathbb{R})$ if $\rho$ sends an
element representing a cusp of $\Gamma$ to a hyperbolic element.

######  Corollary 1.8.

Let $\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{R})$ be a maximal
representation and let $K\subset\Sigma=\Gamma\backslash{\mathcal{H}}^{2}$ be a
compact subset. Then there are constants $0<c_{1}\leq c_{2}$ such that for
every $\gamma\in\Gamma$ representing a closed geodesic $c$ contained in $K$

$c_{1}\ell(c)\leq L(\rho(\gamma))\leq c_{2}\ell(c)\,.$

In particular this holds uniformly for all $\gamma$ representing simple closed
geodesics.

This corollary is well known for Anosov representations. However if $\Sigma$
is not compact, a maximal representation is not necessarily Anosov since the
images of parabolic elements can be unipotent (see for instance §8.3).

### Actions with compatible barycenters

The proof of Theorem 1.5 is carried out in the framework of actions with
compatible barycenters that we now define. Given an isometric $\Gamma$-action
on a metric space $(\mathcal{X},d)$, we say that a map

$\beta\colon X^{(3)}\longrightarrow\mathcal{X}$

from the set $X^{(3)}$ of distinct triples in $X$ to $\mathcal{X}$ is a
barycenter compatible with the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ if $\beta$ is $S_{3}$-invariant,
$\Gamma$-equivariant and for every $(a,b,c,d)\in X^{[4]}$, we have

$[a,b,c,d]=d(\beta(a,b,d),\beta(a,c,d))\,.$

We show then:

######  Theorem 1.9.

Let $X\subset\partial{\mathcal{H}}^{2}$ be a $\Gamma$-invariant non-empty
subset and $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ a positive crossratio
on $X$. Assume that the geodesic current $\mu$ associated by Theorem 1.6 to
the positive crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$
corresponds to a measured lamination. Then for every isometric $\Gamma$-action
on a metric space $\mathcal{X}$ admitting a barycenter compatible with the
crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$, there is an isometric
$\Gamma$-equivariant map

$\mathcal{V}(\mu)\longrightarrow\mathcal{X}.$

We will see that Theorem 1.9 always applies to a framed action of $\Gamma$ on
an $\mathbf{R}$-tree $\mathcal{T}$ if the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ induced by the action is positive
(Proposition 5.8). This crossratio is always positive in the case of the
action on $\mathcal{B}_{1}^{\mathbf{F}}$ induced by a maximal framed
representation in $\operatorname{SL}(2,\mathbf{F})$ (Theorem 8.1).

When $\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$ is a maximal framed
representation, we will use the geometry of the Siegel space to define a
barycenter map associating to every maximal triple
$(\ell_{1},\ell_{2},\ell_{3})$ of Lagrangians a point
$B(\ell_{1},\ell_{2},\ell_{3})\in\mathcal{B}_{n}^{\mathbf{F}}$. Given a
representation $\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$ with
maximal framing $\varphi\colon X\to\mathcal{L}(\mathbf{F}^{2n})$, we will show
that the map

$\beta(a,b,c)=B(\varphi(a),\varphi(b),\varphi(c))$

defines a barycenter compatible with the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\rho}$ previously defined. Thus
Theorem 1.9 applies whenever $\mu_{\rho}$ corresponds to a measured
lamination. Using [BIPP19, Corollary 1.9] we can find a collection of maximal
subsurfaces $\Sigma^{\prime}\subset\Sigma$ such that Theorem 1.9 holds for the
restriction of $\rho$ to $\Sigma^{\prime}$.

### Structure of the paper

In § 2 we discuss preliminaries on geodesic currents and measured laminations.
The new result is Proposition 2.1 that gives an useful 4-point
characterization of measured laminations among geodesic currents that only
involves a dense subset of $\partial{\mathcal{H}}^{2}$. In § 3 we introduce
positive crossratios and the associated periods. In § 4 we construct the
geodesic current associated to such a crossratio. Theorem 1.6 follows directly
combining Propositions 4.3, 4.8 and 4.10, which are proven in this section.
Corollary 1.7 follows from Proposition 4.12. In § 5 we discuss barycenter
maps, and prove Theorem 1.9. In § 6 we review the geometry of the Siegel space
over real closed fields from [BP17], using this we associate to a maximal
framed action on the $\Lambda$-metric space $\mathcal{B}^{\mathbf{F}}_{n}$ a
positive crossratio (Proposition 6.5), as well as a compatible barycenter map
(§ 6.7). In § 7 we prove the results on maximal framed representations:
Theorems 1.2, 1.4, 1.5 and Corollaries 1.3 and 1.8. § 8 collects interesting
examples of maximal framed representations, illustrating various phenomena.

##  2\. On geodesic currents and measured laminations

In this section we recall the notions of geodesic currents and their Bonahon
intersection (§2.1); then we establish a criterion for the support of a
geodesic current to be a geodesic lamination (§2.2); we end by recalling the
definition of the tree $\mathcal{T}(\mu)$ associated to a current $\mu$ of
lamination type in terms of the straight pseudodistance (see [BIPP19, Section
4]) on ${\mathcal{H}}^{2}$ associated to a general current.

Let $\Sigma:=\Gamma\backslash{\mathcal{H}}^{2}$ be a hyperbolic surface of
finite area and denote by ${\rm pr}\colon{\mathcal{H}}^{2}\to\Sigma$ the
covering map. The boundary $\partial{\mathcal{H}}^{2}$ of ${\mathcal{H}}^{2}$
is endowed with the natural cyclic order. For
$(a,b)\in(\partial{\mathcal{H}}^{2})^{2}$ with $a\neq b$, we will denote the
associated _open interval_ in $\partial{\mathcal{H}}^{2}$ by

$I_{(a,b)}:=\\{x\in\partial{\mathcal{H}}^{2}:\,(a,x,b)\text{ is positively
oriented}\\}\,,$

and the _left half open interval_ $I_{(a,b]}$, _rigth half open interval_
$I_{[a,b)}$ and _closed interval_ $I_{[a,b]}$ accordingly, so for example

$I_{[a,b)}=\\{a\\}\cup I_{(a,b)}\,.$

Given a subset $A\subset\partial{\mathcal{H}}^{2}$, we will denote:

$A^{[4]}:=\\{(x,y,z,t)\in A^{4}:\,(x,y,z,t)\text{ is positively
oriented}\\}\,.$

###  2.1. Geodesic currents

A _geodesic current_ is a flip-invariant $\Gamma$-invariant positive Radon
measure on the set of (oriented) geodesics in ${\mathcal{H}}^{2}$, which we
identify with

$(\partial{\mathcal{H}}^{2})^{(2)}:=\\{(x,y)\in(\partial{\mathcal{H}}^{2})^{2}:\,x\neq
y\\}\,.$

Given a (non-oriented) geodesic $c\subset\Sigma$ that is either closed or
joining two cusps, $\delta_{c}$ will be the geodesic current given by

$\delta_{c}:=\frac{1}{2}\sum_{(a,b)}\delta_{(a,b)}$

where we sum on the set of oriented geodesics
$(a,b)\in(\partial{\mathcal{H}}^{2})^{(2)}$ lifting $c$.

Two geodesics
$(a,b),(a^{\prime},b^{\prime})\in(\partial{\mathcal{H}}^{2})^{(2)}$ are
transverse if they intersect in a point. The group
$\operatorname{PSL}(2,\mathbf{R})$, hence $\Gamma$, acts properly on the open
subset
$\mathcal{G}\subset(\partial{\mathcal{H}}^{2})^{(2)}\times(\partial{\mathcal{H}}^{2})^{(2)}$
of transverse pairs of geodesics. The _Bonahon intersection_ $i(\mu,\nu)$ of
two geodesic currents $\mu$ and $\nu$ is the (possibly infinite)
$\mu\times\nu$-measure of any Borel fundamental domain for $\Gamma$ in
$\mathcal{G}$. Note that, when $\nu$ has compact carrier444 We recall that the
_carrier_ of a geodesic current $\mu$ is the closed subset ${\rm
pr}(\bigcup_{g\in\mathrm{supp}(\mu)}g)\subset\Sigma$. The hypothesis that
$\nu$ has compact carrier ensures that $i(\mu,\nu)<\infty$ and is needed in
the proofs of the continuity of the Bonahon intersection., $i(\mu,\nu)$ is
finite. In order to simplify notations, we set

$i(\mu,c):=i(\mu,\delta_{c})$

for all closed geodesic $c\subset\Sigma$.

###  2.2. Measured laminations and $\mu$-short geodesics

We refer to [Mar, §8.3.4] for preliminaries on measured laminations. The
equivalence between (1) and (2) in the next proposition is classical; we
establish that the two conditions are also equivalent to (4), an additional
$4$-point characterization that uses only a dense subset of
$\partial{\mathcal{H}}^{2}$.

######  Proposition 2.1.

Let $\mu$ be a geodesic current and $X$ a dense subset of
$\partial{\mathcal{H}}^{2}$. The following are equivalent.

  1. (1)

$\mathrm{supp}(\mu)$ is a lamination ;

  2. (2)

$i(\mu,\mu)=0$ ;

  3. (3)

$\mu(I_{(d,a]}\times I_{(b,c]})\cdot\mu(I_{(a,b]}\times I_{(c,d]})=0$ for all
$(a,b,c,d)$ in $(\partial{\mathcal{H}}^{2})^{[4]}$;

  4. (4)

$\mu(I_{(d,a)}\times I_{(b,c)})\cdot\mu(I_{(a,b)}\times I_{(c,d)})=0$ for all
$(a,b,c,d)$ in $X^{[4]}$.

Such a current will be called of lamination type.555While in the introduction
we identified with a slight abuse of notation measured laminations with
currents with zero self intersection, we prefer to keep the objects distinct
for the rest of the paper.

###### Proof.

We first show (1) implies (2) : if $i(\mu,\mu)>0$ then there exist open
subsets $A,B$ of $(\partial{\mathcal{H}}^{2})^{(2)}$ with $A\times
B\subset\mathcal{G}$ and

$(\mu\times\mu)(A\times B)=\mu(A)\mu(B)>0,$

hence $\mu(A),\mu(B)>0$. Then there exists $g\in A\cap\mathrm{supp}(\mu)$ and
$g^{\prime}\in B\cap\mathrm{supp}(\mu)$. Since $A\times B\subset\mathcal{G}$,
in particular $g$ and $g^{\prime}$ intersect in a point, hence
$\mathrm{supp}(\mu)$ is not a lamination.

We now prove that (2) implies (3) : since $i(\mu,\mu)=0$, we have
$(\mu\times\mu)(A\times B)=0$ for all transverse Borel subsets $A,B$ of
$(\partial{\mathcal{H}}^{2})^{(2)}$ (namely every pair of geodesics $(a,b)\in
A$, $(a^{\prime},b^{\prime})\in B$ intersect in one point). The claim follows
as $A=I_{(d,a]}\times I_{(b,c]}$ and $B=I_{(a,b]}\times I_{(c,d]}$ are
transverse.

It is clear that (3) implies (4).

Suppose now (4). Let $g$, $g^{\prime}$ be two geodesics in
$\mathrm{supp}(\mu)$. If they are transverse, then $g=(x,y)$,
$g^{\prime}=(x^{\prime},y^{\prime})$ with $(x,x^{\prime},y,y^{\prime})$
positively oriented. Then by density of $X$ there exists $(a,b,c,d)$ in
$X^{[4]}$ such that $(x,a,x^{\prime},b,y,c,y^{\prime},d)$ is positively
oriented. Then $g\in I_{(d,a)}\times I_{(b,c)}$ and $g^{\prime}\in
I_{(a,b)}\times I_{(c,d)}$, and as $g$, $g^{\prime}$ are in the support of
$\mu$, we have $\mu(I_{(d,a)}\times I_{(b,c)})>0$ and $\mu(I_{(a,b)}\times
I_{(c,d)})>0$, a contradiction. Hence $\mathrm{supp}(\mu)$ is a lamination,
proving (1). ∎

An important concept in [BIPP19] was that of _$\mu$ -short geodesic_, namely a
geodesic not intersecting in a point any geodesic in the support of $\mu$;
observe that a geodesic $(a,b)$ is $\mu$-short if and only if

$\mu(I_{(a,b)}\times I_{(b,a)})=0\,.$

It follows from Proposition 2.1 that if the current $\mu$ is of lamination
type, its support consists of $\mu$-short geodesics.

###  2.3. The tree associated to a current of lamination type

We now recall the construction of the tree $\mathcal{T}(\mu)$ associated to a
current $\mu$ of lamination type. We chose here a description adapted to our
purposes, but this agrees with the standard construction described, for
example, in [MS91] and [Kap09, §11.12].

Given a geodesic current $\mu$, we consider the straight pseudodistance on
${\mathcal{H}}^{2}$ [BIPP19, § 4]

$d_{\mu}(x,y)=\frac{1}{2}\;\big{\\{}\mu(\mathcal{G}^{\pitchfork}_{[x,y)})+\mu(\mathcal{G}^{\pitchfork}_{(x,y]})\big{\\}}$

where for a possibly empty geodesic segment $I\subset{\mathcal{H}}^{2}$ we
define

$\mathcal{G}^{\pitchfork}_{I}=\\{(g_{-},g_{+})\in(\partial{\mathcal{H}}^{2})^{(2)}:|g\cap
I|=1\\}$

as the set of geodesics $g$ that intersect transversely the geodesic segment
$I$.

If $\mu$ is of lamination type, then the quotient metric space
$X_{\mu}={\mathcal{H}}^{2}/\\!\\!\sim$, obtained by identifying points at
$d_{\mu}$-distance zero, is $0$-hyperbolic in the sense of Gromov and can
therefore be canonically embedded in a minimal $\mathbf{R}$-tree
$\mathcal{T}(\mu)$. We will denote by $\mathcal{V}(\mu)$ the image in
$\mathcal{T}(\mu)$ of the complementary regions $\mathcal{R}$ of
$\mathrm{supp}(\mu)$. It corresponds to the set of branching points of
$\mathcal{T}(\mu)$.

Since $\mu$ is $\Gamma$-invariant, the group $\Gamma$ acts on
$\mathcal{T}(\mu)$ and therefore on $\mathcal{V}(\mu)$ by isometries. A direct
consequence of the definition of Bonahon intersection is that, for this
action,

$\ell_{\mathcal{T}(\mu)}(\gamma)=i(\mu,\delta_{c})$

for hyperbolic $\gamma$ representing a closed geodesic $c$.

##  3\. Positive crossratios

In this section we introduce the notion of positive crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$, prove that its periods are well
defined, and discuss examples.

###  3.1. Positive crossratios

Let $\Sigma=\Gamma\backslash{\mathcal{H}}^{2}$ be a finite area hyperbolic
surface and let $\Gamma<\operatorname{PSL}(2,\mathbf{R})$ be its fundamental
group realized as a torsion-free lattice in
$\operatorname{PSL}(2,\mathbf{R})$.

######  Definition 3.1.

Let $X\subset\partial{\mathcal{H}}^{2}$ be a $\Gamma$-invariant non-empty
subset. A crossratio on $X$ is a real valued function
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ defined on $X^{[4]}$ satisfying
the following properties:

  1. (CR1)

it is $\Gamma$-invariant;

  2. (CR2)

$[x,y,z,t]=[z,t,x,y]$ for all $(x,y,z,t)\in X^{[4]}$;

  3. (CR3)

$[x,y,z,t]+[x,z,w,t]=[x,y,w,t]$ whenever $(x,y,z,w,t)$ is positively oriented.

The crossratio is in addition positive if

  1. (CR4)

$[x,y,z,t]\geq 0$ for all $(x,y,z,t)\in X^{[4]}$.

######  Remark 3.2.

There are many different non-equivalent notions of crossratio available in the
literature, and there is no standard choice of the order of the arguments of
the function $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$. More specifically

  * •

if $B(\cdot,\cdot,\cdot,\cdot)$ is a crossratio according to [MZ19, Definition
2.4] (which agrees with [Led95, Definition 1.f]), then

$[a,b,c,d]=B(a,d,c,b)$

is a crossratio according to Definition 3.1. However we do not require
continuity, and our crossratio is defined only on a smaller set.

  * •

if $Cr(\cdot,\cdot,\cdot,\cdot)$ is a crossratio according to [Ham97, p. 1],
then

$[a,b,c,d]=\log Cr(b,c,d,a)$

is a crossratio according to Definition 3.1. However, in [Ham97, p. 1] the
crossratio is Hölder continuous.

  * •

if $\mathbb{B}(\cdot,\cdot,\cdot,\cdot)$ is a crossratio according to [Lab07,
p. 1], then

$[a,b,c,d]=\log\mathbb{B}(a,c,d,b)$

is a crossratio according to Definition 3.1. However, the definition in
[Lab07, p. 1] requires Hölder continuity and a much stronger positivity than
what we impose, namely the strict inequality in (CR4).

As a direct consequence of (CR3), positive crossratios have the following
monotonicity property

######  Lemma 3.3.

For all $(x_{1},x_{2},x_{3},x_{4})$ and
$(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4})$ in $X^{[4]}$
such that

$I_{(x_{4},x_{1})}\subset I_{(x^{\prime}_{4},x^{\prime}_{1})}\quad\text{ and
}\quad I_{(x_{2},x_{3})}\subset I_{(x^{\prime}_{2},x^{\prime}_{3})}\,,$

we have

(2)
$[x_{1},x_{2},x_{3},x_{4}]\leq[x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4}]\,.$

$I_{(x^{\prime}_{4},x^{\prime}_{1})}$$I_{(x^{\prime}_{2},x^{\prime}_{3})}$$x_{3}$$x_{2}$$x_{4}$$x_{1}$$x^{\prime}_{3}$$x^{\prime}_{2}$$x^{\prime}_{4}$$x^{\prime}_{1}$

To gain some intuition on the properties (CR2) and (CR3), we recall that if
$x,y,z,t\in\partial{\mathcal{H}}^{2}=\mathbf{R}\cup\\{\infty\\}$ and

$[x,y,z,t]=\ln\frac{(x-z)(y-t)}{(x-y)(z-t)}$

is the logarithm of the usual crossratio, the Liouville measure $\mathcal{L}$
has the property that

(3) $\mathcal{L}(I_{(t,x)}\times I_{(y,z)})=[x,y,z,t]\,.$

Thus (CR2) corresponds to the flip-invariance of $\mathcal{L}$

$\mathcal{L}(I_{(t,x)}\times I_{(y,z)})=\mathcal{L}(I_{(y,z)}\times
I_{(t,x)})$

and (CR3) to additivity

$\mathcal{L}(I_{(t,x)}\times I_{(y,w)})=\mathcal{L}(I_{(t,x)}\times
I_{(y,z)})+\mathcal{L}(I_{(t,x)}\times I_{(z,w)})$

since

$\mathcal{L}(I_{(t,x)}\times\\{z\\})=0\,.$

$z$$w$$y$$x$$t$

###  3.2. Examples

There are two natural crossratios associated to a geodesic current:

######  Example 3.4.

If $\mu$ is a current, it is easily checked that

$[a,b,c,d]_{\mu}^{+}:=\mu(I_{(d,a]}\times I_{(b,c]})$

defines a positive crossratio on $\partial{\mathcal{H}}^{2}$. Similarly,

$[a,b,c,d]_{\mu}^{-}:=\mu(I_{[d,a)}\times I_{[b,c)})$

defines a positive crossratio on $\partial{\mathcal{H}}^{2}$. Note that these
two crossratios may be different (for example this is the case if
$\mu=\delta_{c}$ for some closed geodesic $c$).

Framed actions on trees give other fundamental examples of crossratios:

######  Example 3.5.

If $\mathcal{T}$ is a real tree, we denote by
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\mathcal{T}}$ the usual crossratio
on the boundary $\partial_{\infty}\mathcal{T}$ of the tree $\mathcal{T}$: for
every pairwise distinct $(a,b,c,d)\in\partial_{\infty}\mathcal{T}^{4}$,
$[a,b,c,d]_{\mathcal{T}}$ is the signed distance, on the oriented geodesic
from $a$ to $d$, from the orthogonal projection $\beta_{\mathcal{T}}(a,b,d)$
of $b$ to the orthogonal projection $\beta_{\mathcal{T}}(a,c,d)$ of $c$. Note
that

(4)
$|[a,b,c,d]_{\mathcal{T}}|=d(\beta_{\mathcal{T}}(a,b,d),\beta_{\mathcal{T}}(a,c,d))$

where $d$ denotes the distance in $\mathcal{T}$.

A framed action of $\Gamma$ on $\mathcal{T}$ is an action by isometries
$\rho\colon\Gamma\to\mathrm{Isom}(\mathcal{T})$ admitting an injective
equivariant map (a framing) $\varphi\colon X\to\partial_{\infty}\mathcal{T}$
where $X$ is some $\Gamma$-invariant non-empty subset of
$\partial{\mathcal{H}}^{2}$. Then the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\mathcal{T}}$ on
$\partial_{\infty}\mathcal{T}$ induces a crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\varphi}$ on $X$ defined by

$[x_{1},x_{2},x_{3},x_{4}]_{\varphi}:=[\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3}),\varphi(x_{4})]_{\mathcal{T}}$

for every $(x_{1},x_{2},x_{3},x_{4})\in X^{[4]}$.

######  Example 3.6.

An example of such situation is given by the $\Gamma$-action on the
$\mathbf{R}$-tree $\mathcal{T}(\mu)$ associated to a current of lamination
type. Let $X\subset\partial{\mathcal{H}}^{2}$ be the set of fixed point of
hyperbolic elements whose axis are transverse to the geodesic lamination
$\mathrm{supp}(\mu)$. Then for every such $\gamma\in\Gamma$ with
$\\{\gamma_{-},\gamma_{+}\\}\subset X$, the element $\gamma$ acts on
$\mathcal{T}(\mu)$ with strictly positive translation length
$\ell_{\mathcal{T}(\mu)}(\gamma)=i(\mu,\delta_{c})$ (see § 2.3) and has thus
an attractive fixed point $\varphi(\gamma_{+})$ and a repulsive one
$\varphi(\gamma_{-})$ in $\partial_{\infty}\mathcal{T}(\mu)$. Then
$\varphi\colon X\to\partial_{\infty}\mathcal{T}(\mu)$ is a framing and it
follows from the definition of the distance on $\mathcal{T}(\mu)$ that

$\mu(I_{[x_{4},x_{1}]}\times
I_{[x_{2},x_{3}]})=[\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3}),\varphi(x_{4})]_{\mathcal{T}}\,.$

It follows from the discussion recalled in §2.2 that the crossratio is
positive.

Example 3.5 inspires the following definition:

######  Definition 3.7.

We say that a crossratio is ultrametric if it satisfies :

(CRU) $[a,b,c,d]\times[b,c,d,a]=0$ for all $(a,b,c,d)$ in $X^{[4]}$.

The following is clear.

######  Proposition 3.8.

The crossratio induced by a framed action on a $\mathbf{R}$-tree is
ultrametric.

The following is a corollary of Proposition 2.1.

######  Proposition 3.9.

The crossratio

$[a,b,c,d]:=\mu(I_{(d,a]}\times I_{(b,c]})$

associated to a lamination type current $\mu$ is ultrametric.

###  3.3. The periods of the crossratio

Let now $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ be a positive crossratio
defined on $X\subset\partial{\mathcal{H}}^{2}$, and let $\gamma\in\Gamma$ be
hyperbolic such that $\\{\gamma_{-},\gamma_{+}\\}\subset X$. The additivity
property (CR3) of the crossratio implies that the value $[\gamma_{-},x,\gamma
x,\gamma_{+}]$ is independent of $x\in I_{(\gamma_{-},\gamma_{+})}\cap X$.
This justifies the following:

######  Definition 3.10.

If $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ is a positive crossratio on $X$
and $\gamma\in\Gamma$ is hyperbolic such that
$\\{\gamma_{-},\gamma_{+}\\}\subset X$, the period of $\gamma$ is defined as

$\mathrm{per}(\gamma):=[\gamma_{-},x,\gamma x,\gamma_{+}]\,.$

for one (any) $x\in I_{(\gamma_{-},\gamma_{+})}\cap X$.

The purpose of this section is to extend the definition of the period of a
crossratio defined on a $\Gamma$-invariant set
$X\subset\partial{\mathcal{H}}^{2}$ to hyperbolic elements $\gamma\in\Gamma$
whose endpoints do not necessarily belong to the set $X$.

This is achieved by the following:

######  Proposition 3.11.

Let $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ be a positive crossratio on
$X$, and $\gamma\in\Gamma$ be a hyperbolic element. Choose monotone sequences
$(x_{n}),(x_{n}^{\prime})\subset X$ with limit $\gamma_{-}$ and
$(y_{n}),(y_{n}^{\prime})\subset X$ with limit $\gamma_{+}$. Assume
furthermore that
$(x_{n}^{\prime},\gamma_{-},x_{n},y_{n},\gamma_{+},y_{n}^{\prime})$ is
positively oriented. Then, for all $x\in X$,

$\lim_{n\to\infty}[x_{n},x,\gamma
x,y_{n}]=\lim_{n\to\infty}[x_{n}^{\prime},x,\gamma x,y_{n}^{\prime}].$

$x_{n}^{\prime}$$\gamma_{-}$$x_{n}$$y_{n}^{\prime}$$\gamma_{+}$$y_{n}$$x$$\gamma
x$ Figure 1. Proposition 3.11

###### Proof.

Up to passing to a subsequence we can and will assume that $(x_{0},x,\gamma
x,y_{0})$ is positive (see Figure 1). Since by (CR2) and (CR3) we have

$[x_{n},x,\gamma x,y_{n}]=[x_{n},x,\gamma
x,x_{n}^{\prime}]+[x_{n}^{\prime},x,\gamma
x,y_{n}^{\prime}]+[y_{n}^{\prime},x,\gamma x,y_{n}],$

it is enough to show that

$\lim_{n\to\infty}[x_{n},x,\gamma x,x_{n}^{\prime}]=0$

and the analogous statement for $[y_{n}^{\prime},x,\gamma x,y_{n}]$.

Since, by (CR4), the crossratio is positive, and
$\gamma^{-n}x_{0}\to\gamma_{-}$, it is in turn enough to show that

$\lim_{n\to\infty}[\gamma^{-n}x_{0},x,\gamma x,\gamma^{-n}x_{0}^{\prime}]=0.$

This follows since, for every $N$,

$\displaystyle\infty>[x_{0},x,y_{0}^{\prime},x_{0}^{\prime}]$
$\displaystyle\geq[x_{0},x,\gamma^{N}x,x_{0}^{\prime}]$
$\displaystyle\geq\sum_{j=0}^{N-1}[x_{0},\gamma^{j}x,\gamma^{j+1}x,x_{0}^{\prime}]$
$\displaystyle=\sum_{j=0}^{N-1}[\gamma^{-j}x_{0},x,\gamma
x,\gamma^{-j}x_{0}^{\prime}].$

Here in the last equality we used that the crossratio is $\Gamma$-invariant.
The claim for $[y_{n}^{\prime},x,\gamma x,y_{n}]$ follows analogously. ∎

Thanks to Proposition 3.11 we can extend Definition 3.10 to

######  Definition 3.12.

If $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ is a positive crossratio on $X$
and $\gamma\in\Gamma$ is hyperbolic, the period of $\gamma$ is

$\mathrm{per}(\gamma):=\lim_{\begin{subarray}{c}s,t\in X\\\ s\to\gamma_{-}\\\
t\to\gamma_{+}\end{subarray}}[s,x,\gamma x,t]$

for one (any) $x\in I_{(\gamma_{-},\gamma_{+})}\cap X$.

##  4\. The geodesic current associated to a positive crossratio

In this section we prove Theorem 1.6 and Corollary 1.7. The proof of Theorem
1.6 is carried out in three steps: in §4.1 we use a crossratio to construct a
geodesic current $\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]}$; in §4.2 we
relate the periods of the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ and the intersection of curves
with $\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]}$; in §4.3 we conclude
the proof of Theorem 1.6 by showing that
$\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]}$ depends continuously on the
crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$. The fact that an
integer valued crossratio leads to a multicurve (Corollary 1.7) is shown in
§4.4. We conclude the section discussing in §4.5 how crossratios and geodesic
currents can be restricted to subsurfaces; this is for future reference and
will be used in the study of the real spectrum compactification of maximal
representations.

###  4.1. Construction of the current

The aim of this section is to show that a positive crossratio always leads to
a geodesic current. This is done in Proposition 4.3. The strategy of the proof
is first to associate to the crossratio a finitely additive set function $r$
defined on the family of proper rectangles with vertices in $X^{[4]}$
(Proposition 4.1), and then to build a canonical Radon measure $\mu$ out of
$r$.

Fix a $\Gamma$-invariant non-empty subset $X\subset\partial{\mathcal{H}}^{2}$,
and a positive crossratio

$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]\colon
X^{[4]}\longrightarrow[0,\infty).$

A rectangle in $(\partial{\mathcal{H}}^{2})^{(2)}$ is the product $R=I\times
J$ of two disjoint intervals $I,J\subset\partial{\mathcal{H}}^{2}$. It is
called proper if its closure in $(\partial{\mathcal{H}}^{2})^{(2)}$ is
compact, that is $\overline{I}\cap\overline{J}=\varnothing$, and $I$ and $J$
have non-empty interior. The vertices of $R$ are then the unique positively
oriented $4$-tuple $(a,b,c,d)$ in $(\partial{\mathcal{H}}^{2})^{[4]}$ such
that $d,a$ are the enpoints of $I$ and $b,c$ are the endpoints of $J$,
equivalently

$I_{(d,a)}\times I_{(b,c)}\subset R\subset I_{[d,a]}\times I_{[b,c]}\;.$

For $\mathcal{A}\subset\partial{\mathcal{H}}^{2}$, we denote
$\mathcal{R}(\mathcal{A})$ the family of all proper rectangles with vertices
in $\mathcal{A}^{[4]}$.

If $R$ is a proper rectangle with vertices $(a,b,c,d)$ in $X^{[4]}$ we define

$r(R)=[a,b,c,d]$

the crossratio of the rectangle $R$. It follows directly from the additivity
property (CR3) of the crossratio that this defines a finitely additive
positive function on the family $\mathcal{R}(X)$ of all proper rectangles with
vertices in $X^{[4]}$:

######  Proposition 4.1.

The function $r\colon\mathcal{R}(X)\to\mathbf{R}$ satisfies the following.

  1. (1)

If a rectangle $R\in\mathcal{R}(X)$ is the union $R=R_{1}\sqcup R_{2}$ of two
rectangles with disjoint interior in $\mathcal{R}(X)$, then

$r(R)=r(R_{1})+r(R_{2}).$

  2. (2)

For all $R,R^{\prime}$ in $\mathcal{R}(X)$, if $R\subset R^{\prime}$ then
$r(R)\leq r(R^{\prime})$ .

######  Remark 4.2.

The function $r\colon\mathcal{R}(X)\to\mathbf{R}$ may not be
$\sigma$-additive, even restricting to the family of left half open rectangles
$I_{(d,a]}\times I_{(b,c]}$ with $(a,b,c,d)\in X^{[4]}$. For example setting

$[a,b,c,d]:=\delta_{e}(I_{[d,a)}\times I_{[b,c)})$

for a closed curve $e$ corresponding to some hyperbolic $\gamma\in\Gamma$, we
get a positive crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ on
$X^{[4]}=\partial{\mathcal{H}}^{2}$ whose associated function $r$ is not
$\sigma$-additive on $\mathcal{R}(X)$. Take $a$,$c$ such that
$(a,\gamma_{+},c,\gamma_{-})\in X^{[4]}$ and $I_{[\gamma_{-},a]}\times
I_{[\gamma_{+},c]}$ contains no other point of the orbit of
$(\gamma_{-},\gamma_{+})$. Let $d_{n}\downarrow\gamma_{-}$ in
$I_{(\gamma_{-},a)}$ and $b_{n}\downarrow\gamma_{+}$ in $I_{(\gamma_{+},c)}$.
Let $R_{n}=I_{(d_{n},a]}\times I_{(b_{n},c]}$. Then
$R=I_{(\gamma_{-},a]}\times I_{(\gamma_{+},c]}$ is the increasing union of the
$R_{n}$, and $r(R_{n})=[a,b_{n},c,d_{n}]=0$ for all $n$ whereas
$r(R)=[a,\gamma_{+},c,\gamma_{-}]=1$, contradicting $\sigma$-additivity. The
problem is due to the fact that this crossratio is not continuous at
$(a,\gamma_{+},c,\gamma_{-})$.

We now construct the mesure $\mu$. Recall that for a rectangle
$R\in\mathcal{R}(X)$ with vertices $(a,b,c,d)$ we set $r(R)=[a,b,c,d]$.
Furthermore we denote by $\mathring{R}$ (resp. $\overline{R}$) the open (resp.
closed) rectangle with the same vertices as $R$.

######  Proposition 4.3.

There exists a unique positive Radon measure $\mu$ on
$(\partial{\mathcal{H}}^{2})^{(2)}$ satisfying one of the following equivalent
conditions.

  1. (1)

$\mu(\mathring{R})\leq r(R)\leq\mu(\overline{R})$ for any (proper) rectangle
$R\in\mathcal{R}(X)$.

  2. (2)

For any (proper) rectangles $R,R^{\prime}\in\mathcal{R}(X)$ with
$\overline{R^{\prime}}\subset\mathring{R}$, we have $\mu(R^{\prime})\leq r(R)$
and $r(R^{\prime})\leq\mu(R)$.

  3. (3)

For all (proper) open rectangles $R\in\mathcal{R}(\partial{\mathcal{H}}^{2})$

$\mu(R)=\sup\left\\{r(R^{\prime}):\,R^{\prime}\in\mathcal{R}(X)\text{ and
}\overline{R^{\prime}}\subset R\right\\}.$

  4. (4)

For all (proper) closed rectangles
$R\in\mathcal{R}(\partial{\mathcal{H}}^{2})$

$\mu(R)=\inf\left\\{r(R^{\prime}):\,R^{\prime}\in\mathcal{R}(X)\text{ and
}R\subset\mathring{R}^{\prime}\right\\}.$

######  Definition 4.4.

We call the measure $\mu$ in Proposition 4.3 the geodesic current associated
to the positive crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]\colon
X^{[4]}\to\mathbf{R}$, and denote it by
$\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]}$ if we want to emphasize the
dependence on $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$.

Proposition 4.3 implies the following “outer and inner” continuity properties
of the current.

######  Proposition 4.5.

Let $(a,b,c,d)$ be a positively oriented quadruple in
$\partial{\mathcal{H}}^{2}$. Let $(a_{n},b_{n},c_{n},d_{n})_{n\geq 1}$ be a
sequence in $X^{[4]}$ converging to $(a,b,c,d)$. Then

  1. (1)

If $d_{n},a_{n}\in I_{(d,a)}$ and $b_{n},c_{n}\in I_{(b,c)}$ for all $n\geq
1$,
then $\mu(I_{(d,a)}\times I_{(b,c)})=\lim_{n}[a_{n},b_{n},c_{n},d_{n}]$;

  2. (2)

If $a_{n},b_{n}\in I_{(a,b)}$ and $c_{n},d_{n}\in I_{(c,d)}$ for all $n\geq
1$,
then $\mu(I_{[d,a]}\times I_{[b,c]})=\lim_{n}[a_{n},b_{n},c_{n},d_{n}]$.

###### Proof.

We prove the first assertion (the second is similar). We have by (3)

(5) $\mu(I_{(d,a)}\times
I_{(b,c)})=\sup\left\\{[a^{\prime},b^{\prime},c^{\prime},d^{\prime}]\left|\,\begin{array}[]{l}a^{\prime},d^{\prime}\in
I_{(d,a)},\\\ b^{\prime},c^{\prime}\in I_{(b,c)}\text{ and }\\\
(a^{\prime},b^{\prime},c^{\prime},d^{\prime})\in
X^{[4]}\end{array}\right.\right\\}\,,$

Let $(a^{\prime},b^{\prime},c^{\prime},d^{\prime})\in\mathcal{X}^{[4]}$ with
$a^{\prime},d^{\prime}\in I_{(d,a)}$ and $b^{\prime},c^{\prime}\in I_{(b,c)}$.
For $n$ large enough we have

$I_{[d^{\prime},a^{\prime}]}\subset I_{[d_{n},a_{n}]}\qquad\text{ and }\qquad
I_{[b^{\prime},c^{\prime}]}\subset I_{[b_{n},c_{n}]}\,,$

and hence

$[a^{\prime},b^{\prime},c^{\prime},d^{\prime}]\leq[a_{n},b_{n},c_{n},d_{n}]\,,$

which by (5) implies (1). ∎

###### Proof of Proposition 4.3.

We begin by proving that the conditions are equivalent. Let $\mu$ be any
positive Radon measure on $(\partial{\mathcal{H}}^{2})^{(2)}$.

It is clear that (1) implies (2).

We now prove that (2) implies (3). Consider a open rectangle $R$ in
$\mathcal{R}(\partial{\mathcal{H}}^{2})$. First observe that for every
$R^{\prime}$ in $\mathcal{R}(X)$ such that $\overline{R^{\prime}}\subset R$,
we have by (2) that $r(R^{\prime})\leq\mu(R)$, hence

$\sup\left\\{r(R^{\prime}):\,R^{\prime}\in\mathcal{R}(X)\text{ and
}\overline{R^{\prime}}\subset R\right\\}\leq\mu(R)\;.$

By density of $X$ in $\partial{\mathcal{H}}^{2}$ we can now take an increasing
sequence of rectangles $R_{n}$ in $\mathcal{R}(X)$ with union $R$ such that
$\overline{R_{n}}\subset\mathring{R}_{n+1}$. We have by (2) that

$\mu(R_{n-1})\leq r(R_{n})\leq\mu(R_{n+1})$

in particular by $\sigma$-additivity of $\mu$ we have
$\mu(R)=\lim_{n}\mu(R_{n})=\lim_{n}r(R_{n})$.

We now prove that (3) implies (4). Consider a proper closed rectangle $R$ in
$\partial{\mathcal{H}}^{2}$. For every $R^{\prime\prime}$ in $\mathcal{R}(X)$
such that $R\subset\mathring{R}^{\prime\prime}$, there is an open $R^{\prime}$
in $\mathcal{R}(X)$ such that $R\subset R^{\prime}$ and
$\overline{R^{\prime}}\subset\mathring{R}^{\prime\prime}$. Then we have
$\mu(R)\leq\mu(R^{\prime})\leq r(R^{\prime\prime})$ by (3). Hence

$\mu(R)\leq\inf\left\\{r(R^{\prime\prime}):\,R^{\prime\prime}\in\mathcal{R}(X)\text{
and }R\subset\mathring{R}^{\prime\prime}\right\\}\;.$

Let now $R_{n}$ be a decreasing sequence of open rectangles in
$\mathcal{R}(X)$ with intersection $R$, such that $\overline{R_{n+1}}\subset
R_{n}$. Then $\mu(R)=\lim_{n}\mu(R_{n})$, and by (3) we have
$r(R_{n})\leq\mu(R_{n-1})$ and

$\mu(R_{n})=\sup\left\\{r(R^{\prime}):\,R^{\prime}\in\mathcal{R}(X)\text{ and
}\overline{R^{\prime}}\subset R_{n}\right\\}\leq r(R_{n})$

hence $\mu(R)=\lim_{n}r(R_{n})$.

We finally check that (4) implies (1). Consider any rectangle $R$ in
$\mathcal{R}(X)$. As $r(R)\leq r(R^{\prime})$ for all $R^{\prime}$ containing
$R$, taking infimum on $R^{\prime}$ containing $\overline{R}$ in their
interior we get by (4) that $r(R)\leq\mu(\overline{R})$. Now write the open
rectangle $\mathring{R}$ as a increasing union $\mathring{R}=\cup\uparrow
R_{n}$ of closed rectangles $R_{n}$ in $\mathcal{R}(X)$ with
$R_{n}\subset\mathring{R}_{n+1}$. Then by (4) we have $\mu(R_{n})\leq
r(R_{n+1})\leq r(R)$. As $\mu(R_{n})\to\mu(\mathring{R})$ by
$\sigma$-additivity, we deduce $\mu(\mathring{R})\leq r(R)$.

We now prove the existence of $\mu$ satisfying (2). The strategy of the
construction of $\mu$ is to use the finitely additive function $r$ to define
the integral of compactly supported continuous functions. This leads by the
Riesz representation theorem to a Radon measure $\mu$.

A simple function is a linear combination
$g=\sum_{i=1}^{n}\alpha_{i}\chi_{R_{i}}$ of characteristic functions of
rectangles $R_{i}$ in $\mathcal{R}(X)$. Define $E(g)$ by

$E(g):=\sum_{i=1}^{n}\alpha_{i}r(R_{i})\,.$

The additivity property of $r$ on $\mathcal{R}(X)$ (Proposition 4.1) shows
that $E(g)$ is independent of the representation of $g$ as linear combination
of characteristic functions of proper rectangles in $\mathcal{R}(X)$. It
implies that if $g_{1}$ and $g_{2}$ are simple functions, then

$E(g_{1}+g_{2})=E(g_{1})+E(g_{2})\,.$

This property also shows that if $g_{1},g_{2}$ are simple and $g_{1}\leq
g_{2}$, then $E(g_{1})\leq E(g_{2})$.

If now $f\geq 0$ is a continuous function on
$(\partial{\mathcal{H}}^{2})^{(2)}$ with compact support, define

$I_{+}(f):=\sup\\{E(g):0\leq g\leq f,\,g\text{ is simple}\\}$

and

$I_{-}(f):=\inf\\{E(g):f\leq g,\,g\text{ is simple}\\}\;.$

Then by uniform continuity of $f$ and density of $X$, we have
$I_{-}(f)=I_{+}(f)=:I(f)$. The additivity of $I$ on positive continuous
functions with compact support then follows from the fact that $I_{-}$ is
super-additive, and $I_{+}$ is subadditive. Then $I$ extends to all continuous
functions with compact support as a positive linear functional on the space of
continuous functions with compact support, hence corresponds to a Radon
measure $\mu$.

We now prove that $\mu$ satisfies (2). Let $R$, $R^{\prime}$ be rectangles
with $\overline{R^{\prime}}\subset\mathring{R}$. As there is a continuous
function $f$ with compact support such that $\chi_{R^{\prime}}\leq
f\leq\chi_{R}$, we have $\mu(R^{\prime})\leq I(f)\leq E(\chi_{R})=r(R)$
whenever $R\in\mathcal{R}(X)$, and $r(R^{\prime})=E(\chi_{R^{\prime}})\leq
I(f)\leq\mu(R)$ whenever $R^{\prime}\in\mathcal{R}(X)$.

Uniqueness comes from (3), as the class of proper open rectangles in
$(\partial{\mathcal{H}}^{2})^{(2)}$ is stable under finite intersection and
generates the Borel $\sigma$-algebra. ∎

######  Remark 4.6.

It follows from Proposition 4.3 (4) that also for all pencils $P=\\{a\\}\times
I_{[b,c]}$,

$\mu(P)=\inf\left\\{r(R^{\prime}):\,R^{\prime}\in\mathcal{R}(X)\text{ and
}P\subset\mathring{R}^{\prime}\right\\}.$

######  Proposition 4.7.

If a positive crossratio is ultrametric, then the associated current $\mu$ is
of lamination type.

###### Proof.

By Proposition 2.1 it is enough to prove that, for all
$(a,b,c,d)\in{X}^{[4]}$, $\mu(I_{(d,a)}\times
I_{(b,c)})\cdot\mu(I_{(a,b)}\times I_{(c,d)})=0$.

Let $(a,b,c,d)$ in ${X}^{[4]}$. As the crossratio is ultrametric, we have
either $[a,b,c,d]=0$ or $[b,c,d,a]=0$. As $\mu(I_{(d,a)}\times
I_{(b,c)})\leq[a,b,c,d]$ and $\mu(I_{(a,b)}\times I_{(c,d)})\leq[b,c,d,a]$
(see Proposition 4.3(1)), this implies that $\mu(I_{(d,a)}\times I_{(b,c)})=0$
or $\mu(I_{(a,b)}\times I_{(c,d)})=0$. ∎

###  4.2. Periods and intersections

We now turn to the problem of identifying the periods of a positive crossratio
with the intersections of the corresponding current. Let then
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ be a positive crossratio defined
on $X$, and let $\gamma\in\Gamma$ be hyperbolic. Recall from §3.3 the
definition of the period of $\gamma$ with respect to the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$.

In the following proposition we will use the well known fact that if $\gamma$
is a hyperbolic element representing a closed geodesic $c\subset\Sigma$, and
$\mu$ is a geodesic current, since $I_{(\gamma_{+},\gamma_{-})}\times
I_{(x,\gamma x]}$ is a Borel fundamental domain for the
$\langle\gamma\rangle$-action on $I_{(\gamma_{+},\gamma_{-})}\times
I_{(\gamma_{-},\gamma_{+})}$, the intersection $i(\mu,\delta_{c})$ can be
computed as

$i(\mu,\delta_{c})=\mu(I_{(\gamma_{+},\gamma_{-})}\times I_{(x,\gamma x]})\,.$

######  Proposition 4.8.

Let $\mu$ be the geodesic current associated to a positive crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ and let $c$ be a closed geodesic
represented by an hyperbolic element $\gamma\in\Gamma$. Then

$\mathrm{per}(\gamma)=i(\mu,\delta_{c})\,.$

###### Proof.

In the notation of Proposition 3.11 (see also Figure 1) we have that

(6) $\mathrm{per}(\gamma)=\lim_{n\to\infty}[x_{n},x,\gamma x,y_{n}]$

and also

(7) $\mathrm{per}(\gamma)=\lim_{n\to\infty}[x^{\prime}_{n},x,\gamma
x,y^{\prime}_{n}]\,,$

where $x\in I_{(\gamma_{-},\gamma_{+})}\cap X$ is arbitrary.

For any $x\in I_{(\gamma_{-},\gamma_{+})}\cap X$ and $n\in\mathbf{N}$ we have

$\displaystyle i(\mu,\delta_{c})$
$\displaystyle=\mu(I_{(\gamma_{+},\gamma_{-})}\times I_{(x,\gamma x]})$
$\displaystyle=\mu(I_{(\gamma_{+},\gamma_{-})}\times I_{(x,\gamma
x)})+\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{\gamma x\\})$
$\displaystyle\leq[x_{n},x,\gamma
x,y_{n}]+\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})\,,$

where the inequality follows from Proposition 4.3 (1). By (6) this implies
that

$i(\mu,\delta_{c})\leq\mathrm{per}(\gamma)+\mu(I_{(\gamma_{-},\gamma_{+})}\times\\{x\\})\,.$

Next we have:

$i(\mu,\delta_{c})=\mu(I_{(\gamma_{+},\gamma_{-})}\times I_{(x,\gamma
x]})=\mu(I_{(\gamma_{+},\gamma_{-})}\times I_{[x,\gamma
x]})-\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})\,.$

Using Lemma 4.9 below this equals

$\mu(I_{[\gamma_{+},\gamma_{-}]}\times I_{[x,\gamma
x]})-\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})\,,$

which, again by Proposition 4.3 (1), implies

$\displaystyle i(\mu,\delta_{c})=$
$\displaystyle\mu(I_{[\gamma_{+},\gamma_{-}]}\times I_{[x,\gamma
x]})-\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})$ $\displaystyle\geq$
$\displaystyle[x^{\prime}_{n},x,\gamma
x,y^{\prime}_{n}]-\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})\,.$

Then it follows from (7) that

$i(\mu,\delta_{c})\geq\mathrm{per}(\gamma)-\mu(I_{(\gamma_{-},\gamma_{+})}\times\\{x\\})$

and thus

$|i(\mu,\delta_{c})-\mathrm{per}(\gamma)|\leq\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})$

for any $x\in I_{(\gamma_{-},\gamma_{+})}\cap X$.

Now fix a closed interval $I_{[a,b]}\subset I_{(\gamma_{-},\gamma_{+})}$ with
non-empty interior. Then

$\sum_{x\in
X\cap[a,b]}\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x\\})\leq\mu(I_{(\gamma_{+},\gamma_{-})}\times
I_{[a,b]})<\infty$

and since $X\cap[a,b]$ is infinite, this implies the existence of a sequence
$(x_{n})_{n\geq 1}$ in $X\cap[a,b]$ with

$\lim_{n\to\infty}\mu(I_{(\gamma_{+},\gamma_{-})}\times\\{x_{n}\\})=0\,,$

which implies that $i(\mu,\delta_{c})=\mathrm{per}(\gamma)$. ∎

###  4.3. The current depends continuously on the crossratio

The vector space $\mathcal{CR}(X)$ of crossratios on $X$ is a topological
vector space for the topology of pointwise convergence and the space
$\mathcal{CR}^{+}(X)$ of positive crossratios is a closed convex cone in it.
We observe moreover that the map

$\displaystyle\mathcal{CR}^{+}(X)$
$\displaystyle\longrightarrow\mathcal{C}(\Sigma)$
$\displaystyle[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$
$\displaystyle\mapsto\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]}$

from positive crossratios to the space $\mathcal{C}(\Sigma)$ of geodesic
currents is surjective. In fact, if $\mu$ is a geodesic current, one verifies
using the regularity of $\mu$ that
$\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\mu}^{+}}=\mu$ for the
crossratio $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\mu}^{+}$ of Example
3.4.

Let ${H_{\Gamma}}$ denote the subset of $\partial{\mathcal{H}}^{2}$ consisting
of the fixed points of hyperbolic elements in $\Gamma$. Recall that if
$\gamma\in\Gamma$ is hyperbolic we let $\gamma_{+}$ and $\gamma_{-}$ denote
respectively the attractive and the repulsive fixed point of $\gamma$. For
every $a\in{H_{\Gamma}}$, we choose $\gamma$ such that $a=\gamma_{-}$ and we
denote by $\overline{a}$ the point $\gamma_{+}$.

The following simple lemma is crucial:

######  Lemma 4.9 ([Mar, Proposition 8.2.8]).

Let $\mu$ be a geodesic current, $a\in{H_{\Gamma}}$ and
$I\subset\partial{\mathcal{H}}^{2}$ be a closed interval with
$\\{a,\overline{a}\\}\cap I=\varnothing$. Then $\mu(\\{a\\}\times I)=0$.

We call a $\Gamma$-invariant subset $S_{\Gamma}\subset{H_{\Gamma}}$ symmetric
if $\overline{\xi}\in S_{\Gamma}$ whenever $\xi\in S_{\Gamma}$.

######  Proposition 4.10.

Let $S_{\Gamma}\subset{H_{\Gamma}}$ be a $\Gamma$-invariant symmetric subset
such that $\\{(\xi,\overline{\xi}):\xi\in S_{\Gamma}\\}$ is dense in
$(\partial{\mathcal{H}}^{2})^{2}$. Then the map

$\displaystyle\mathcal{CR}^{+}(S_{\Gamma})$
$\displaystyle\longrightarrow\mathcal{C}(\Sigma)$
$\displaystyle[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$
$\displaystyle\mapsto\mu_{[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]}$

is continuous.

Observe that Proposition 4.10 applies to $S_{\Gamma}={H_{\Gamma}}$ in
particular. In the proof of Proposition 4.10 we will focus on a special subset
of $S_{\Gamma}^{[4]}$: we say that a quadruple $(a,b,c,d)\in S_{\Gamma}^{[4]}$
is good if $\\{\overline{a},\overline{d}\\}\cap I_{[b,c]}=\varnothing$ and
$\\{\overline{b},\overline{c}\\}\cap I_{[d,a]}=\varnothing$.

######  Lemma 4.11.

Let $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]\in\mathcal{CR}^{+}(S_{\Gamma})$
and let $\mu$ be the associated current. For every good quadruple
$(a,b,c,d)\in S_{\Gamma}^{[4]}$ we have

$\mu(I_{(d,a)}\times I_{(b,c)})=\mu(I_{[d,a]}\times I_{[b,c]})=[a,b,c,d]\,.$

###### Proof.

This follows immediately from the inequalities in Proposition 4.3 (1) and
Lemma 4.9 applied to $\\{d\\}\times I_{[b,c]}$, $\\{a\\}\times I_{[b,c]}$,
$I_{[d,a]}\times\\{b\\}$ and $I_{[d,a]}\times\\{c\\}$. ∎

###### Proof of Proposition 4.10.

Let $[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{n}$ and
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ be positive crossratios on
$S_{\Gamma}$ and $\mu_{n}$, $\mu$ the associated geodesic currents. Assume
that

$\lim_{n}[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{n}=[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]\,.$

We have to show that for every positive continuous function $f$ on
$(\partial{\mathcal{H}}^{2})^{(2)}$ with compact support,
$\lim_{n\to\infty}\mu_{n}(f)=\mu(f)$. Using a finite partition of unity we may
assume that the support $\mathrm{supp}(f)$ is contained in some rectangle
$I_{[d,a]}\times I_{[b,c]}$ with $(a,b,c,d)$ positive.

Fix an interval $I_{[a_{0},b_{0}]}\subset I_{(a,b)}$ and observe that for
every open set $J\subset I_{(b_{0},a_{0})}$ the set

$S_{J}:=\\{\xi\in J\cap S_{\Gamma}:\,\overline{\xi}\in I_{[a_{0},b_{0}]}\\}$

is dense in $J$. Choose then $a^{\prime}\in S_{(a,a_{0})}$, $b^{\prime}\in
S_{(b_{0},b)}$, $\\{c^{\prime},d^{\prime}\\}\subset S_{(c,d)}$ with
$(a^{\prime},b^{\prime},c^{\prime},d^{\prime})$ positive and observe that the
quadruple $(a^{\prime},b^{\prime},c^{\prime},d^{\prime})$ is good.

$a$$a^{\prime}$$a_{0}$$b_{0}$$b^{\prime}$$b$$c$$c^{\prime}$$d^{\prime}$$d$

Fix some distance inducing the topology on
$(\partial{\mathcal{H}}^{2})^{(2)}$, fix $\epsilon>0$ and let $\delta>0$ be
such that if $R\subset(\partial{\mathcal{H}}^{2})^{(2)}$ is any closed
rectangle of diameter $\operatorname{diam}(R)<\delta$, then

$\max_{R}f-\min_{R}f<\epsilon\,.$

Fix now a cover of $\mathrm{supp}(f)$ by closed rectangles
$R_{i}=I_{[x_{i},y_{i}]}\times I_{[z_{i},w_{i}]}$, for $i=1,2,\dots,N$, such
that

  1. (1)

$\operatorname{diam}(R_{i})<\delta$;

  2. (2)

the interiors of the rectangles are pairwise disjoint;

  3. (3)

$R_{i}\subset I_{(d^{\prime},a^{\prime})}\times I_{(b^{\prime},c^{\prime})};$

  4. (4)

$\\{x_{i},y_{i}\\}\subset S_{(d^{\prime},a^{\prime})}$ and
$\\{z_{i},w_{i}\\}\subset S_{(b^{\prime},c^{\prime})}$.

Observe that for every $1\leq i\leq N$, the quadruple
$(x_{i},z_{i},w_{i},y_{i})$ is good. As a result, we have (see Lemma 4.11)

  1. (5)

$\mu(R_{i})=[x_{i},z_{i},w_{i},y_{i}],\quad\mu_{n}(R_{i})=[x_{i},z_{i},w_{i},y_{i}]_{n}$,
and

  2. (6)

$\mu(R_{i}\cap R_{j})=0,\quad\mu_{n}(R_{i}\cap R_{j})=0$ for all $i\neq j$.

It follows then that

$\displaystyle\left|\int
f\,d\mu_{n}-\sum_{i=1}^{N}(\min_{R_{i}}f)\mu_{n}(R_{i})\right|$
$\displaystyle\leq\sum_{i=1}^{N}\epsilon\mu_{n}(R_{i})$
$\displaystyle\leq\epsilon\mu_{n}(I_{[d^{\prime},a^{\prime}]}\times
I_{[b^{\prime},c^{\prime}]})$
$\displaystyle=\epsilon[a^{\prime},b^{\prime},c^{\prime},d^{\prime}]_{n}$

and similarly

$\displaystyle\left|\int
f\,d\mu-\sum_{i=1}^{N}(\min_{R_{i}}f)\mu(R_{i})\right|\leq\epsilon[a^{\prime},b^{\prime},c^{\prime},d^{\prime}]\,.$

From these inequalities, the assumption that
$\lim_{n}[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{n}=[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$
and (5), we deduce that

$\overline{\lim_{n\to\infty}}\left|\int f\,d\mu-\int f\,d\mu_{n}\right|\leq
2\epsilon[a^{\prime},b^{\prime},c^{\prime},d^{\prime}]$

and hence

$\lim_{n\to\infty}\int f\,d\mu_{n}=\int f\,d\mu\,.$

∎

###  4.4. Integral crossratios

The goal of the section is to prove the following

######  Proposition 4.12.

If the positive crossratio

$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]\colon X^{[4]}\to[0,\infty)$

takes values in the integers $\mathbf{N}=\\{0,1,2,\ldots\\}$, then the
associated current corresponds to an integral geodesic multicurve.

###### Proof.

We first show that $\mu$ is purely atomic. It follows from Proposition 4.3 (3)
and (4) that $\mu(R)\in\mathbf{N}$ for all proper open and proper closed
rectangles. Let
$g\in\mathrm{supp}(\mu)\subset(\partial{\mathcal{H}}^{2})^{(2)}$ and let
$R_{n}$ be a decreasing sequence of open rectangles with $\bigcap
R_{n}=\\{g\\}$. Since $\mu(R_{n})\in\mathbf{N}$ we have either

  1. (1)

$\mu(R_{n})\geq 1$ for all $n\geq 1$ and $\mu(\\{g\\})\geq 1$

or

  1. (2)

there exists $n_{0}$ such that $\mu(R_{n})=0$ for all $n\geq n_{0}$.

The second case cannot happen since $g$ is in the support of $\mu$. As a
result $\mu$ is purely atomic with $\mathbf{N}$-valued atoms.

We now show that all geodesics in the support of $\mu$ are either closed or
connect two cusps. Let $(a,b)\in(\partial{\mathcal{H}}^{2})^{(2)}$ be such an
atom. Then $\Gamma\cdot(a,b)$ meets every compact subset of
$(\partial{\mathcal{H}}^{2})^{(2)}$ in only finitely many points. As a result,
if $g\subset{\mathcal{H}}^{2}$ is the geodesic connecting $(a,b)$, ${\rm
pr}(g)\subset\Sigma$ is a closed subset where, as always, ${\rm
pr}\colon{\mathcal{H}}^{2}\to\Sigma$ denotes the universal covering map. Thus
either $g$ corresponds to a periodic geodesic, or ${\rm pr}(g)$ is a geodesic
connecting two cusps.

There is a compact subset $K\subset\Sigma$ such that every biinfinite
geodesic, as well as every closed geodesic, meets $K$. Thus if $\mathcal{A}$
denotes the set of atoms of $\mu$ there is a compact subset
$C\subset(\partial{\mathcal{H}}^{2})^{(2)}$ such that for all
$a\in\mathcal{A}$, $\Gamma\cdot a\cap C\neq\emptyset$. This implies that

$\mu=\sum_{c\in F}n_{c}\delta_{c}$

where $F$ is a finite set of geodesics either periodic or connecting to cusps,
$\delta_{c}$ is the geodesic current corresponding to $c$, and
$n_{c}\in\mathbf{N}^{*}$. ∎

###  4.5. Restriction to a subsurface

We conclude the section discussing how the construction of the geodesic
current associated to a positive crossratio behaves with respect to
restriction to subsurfaces. This will be useful in the study of maximal
representations.

Let $\Sigma^{\prime}\subset\Sigma$ be a subsurface with geodesic boundary. Let
$\mathcal{G}(\Sigma^{\prime})\subset(\partial{\mathcal{H}}^{2})^{(2)}$ be the
set of geodesics whose projection lies in the interior
$\mathring{\Sigma}^{\prime}$ of $\Sigma^{\prime}$. If $\mu$ is a current on
$\Sigma$, we define $\mu_{|\Sigma^{\prime}}\in\mathcal{C}(\Sigma)$ by

$\mu_{|\Sigma^{\prime}}:=\chi_{\mathcal{G}(\Sigma^{\prime})}\mu\,,$

where $\chi_{\mathcal{G}(\Sigma^{\prime})}$ is the characteristic function of
$\mathcal{G}(\Sigma^{\prime})$.

We write $i(\mu,\partial\Sigma^{\prime})=0$ when $i(\mu,c)=0$ for every
boundary component $c$ of $\Sigma^{\prime}$. This is the case precisely when
no geodesic in the support of $\mu$ intersects $\partial\Sigma^{\prime}$; thus
in that case we have

$i(\mu,c)=i(\mu_{|\Sigma^{\prime}},c)$

for every closed geodesic $c$ contained in $\Sigma^{\prime}$.

We now choose a finite area hyperbolization
$\Sigma_{0}=\Gamma_{0}\backslash{\mathcal{H}}^{2}$ of
$\mathring{\Sigma}^{\prime}$ and a corresponding identification
$h\colon\Gamma_{0}\to\pi_{1}(\Sigma^{\prime})<\Gamma$. We denote by
$\phi\colon\partial{\mathcal{H}}^{2}\to\partial{\mathcal{H}}^{2}$ an
injective, monotone $h$-equivariant map. This is a quasi-conjugacy that opens
all the cusps corresponding to geodesic boundary components of
$\Sigma^{\prime}$. It follows from this discussion that:

######  Proposition 4.13.

Let $\mu$ be a current on $\Sigma$. Then

$\mu_{0}(A):=\mu((\phi\times\phi)(A))$

defines a current $\mu_{0}=\phi^{*}\mu$ on $\Sigma_{0}$, that we will call the
current induced by $\mu$ on $\Sigma_{0}$. We have the following properties :

  1. (1)

$\phi^{*}\mu=\phi^{*}\mu_{|\Sigma^{\prime}}$ .

  2. (2)

If $\mu_{|\Sigma^{\prime}}^{\prime}$ has compact carrier included in
$\mathring{\Sigma}^{\prime}$, then $\mu_{0}$ has compact carrier in
$\Sigma_{0}$ .

  3. (3)

Let $\mu$, $\nu$ be currents on $\Sigma$. Assume that $\mu_{|\Sigma^{\prime}}$
and $\nu_{|\Sigma^{\prime}}$ have compact carrier included in
$\mathring{\Sigma}^{\prime}$. Then

$i(\mu_{|\Sigma^{\prime}},\nu_{|\Sigma^{\prime}})=i(\mu_{0},\nu_{0}).$

  4. (4)

We have

$i(\mu_{0},\gamma)=i(\mu_{|\Sigma^{\prime}},h(\gamma))$

for all hyperbolic $\gamma\in\Gamma_{0}$. In particular if
$i(\mu,\partial\Sigma^{\prime})=0$ then $i(\mu_{0},\gamma)=i(\mu,h(\gamma))$
for all hyperbolic $\gamma\in\Gamma_{0}$.

  5. (5)

Assume that $i(\mu,\partial\Sigma^{\prime})=0$. If $\mu$ is the current
associated to a positive crossratio $b\in\mathcal{CR}({H_{\Gamma}})$, then
$\mu_{0}$ is the current associated to the positive crossratio
$b_{0}=\phi^{*}b$ in $\mathcal{CR}(H_{\Gamma_{0}})$.

##  5\. Equivariant tree embeddings

In this section we discuss barycenter maps compatible with positive
crossratios and prove Theorem 1.9. In §5.2 we discuss a first class of actions
to which Theorem 1.9 applies: framed actions on trees.

###  5.1. Actions with compatible crossratio and barycenter

Let $\rho\colon\Gamma\to\mathrm{Isom}(\mathcal{X})$ be an action by isometries
on a metric space $(\mathcal{X},d_{\mathcal{X}})$ and
$X\subset\partial{\mathcal{H}}^{2}$ be a non-empty $\Gamma$-invariant subset.

######  Definition 5.1.

  1. (1)

A $\mathcal{X}$-valued barycenter map on $X$ (or just a barycenter map if the
context is clear) is a map

$\beta\colon X^{(3)}\longrightarrow\mathcal{X}$

defined on the set $X^{(3)}$ of triples of distinct points in $X$ that
verifies:

    1. (a)

$\beta$ is $S_{3}$-invariant;

    2. (b)

$\beta$ is $\Gamma$-equivariant.

  2. (2)

A barycenter map is compatible with a positive crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]$ defined on $X$ if, whenever
$(a,b,c,d)\in X^{[4]}$,

$[a,b,c,d]=d_{\mathcal{X}}(\beta(a,b,d),\beta(a,c,d))\,.$

Condition (2) is inspired by the construction of a crossratio induced by a
framed action on a tree, as in Example 3.5. Indeed we have:

######  Example 5.2.

Let $\rho\colon\Gamma\to\mathrm{Isom}(\mathcal{T})$ be a framed action of
$\Gamma$ on a real tree $\mathcal{T}$ with framing $\varphi\colon
X\to\partial_{\infty}\mathcal{T}$. Then the usual barycenter
$B\colon\partial_{\infty}\mathcal{T}^{(3)}\to\mathcal{T}$ induces a barycenter
map on $X$ compatible with the positive crossratio
$[\,\cdot\,,\,\cdot\,,\cdot\,,\,\cdot\,]_{\varphi}$.

The goal of the section is to prove the following:

######  Theorem 5.3.

Let $\rho\colon\Gamma\to\mathrm{Isom}(\mathcal{X})$ be an isometric action.
Assume that there is a positive crossratio on $X$ and a compatible barycenter
map, that the geodesic current $\mu$ associated to the positive crossratio is
of lamination type and let $\mathcal{V}(\mu)$ be the set of vertices of the
$\mathbf{R}$-tree $\mathcal{T}(\mu)$ associated to $\mu$. Then there is an
equivariant isometric embedding

$\mathcal{V}(\mu)\hookrightarrow\mathcal{X}\,.$

In order to define the embedding, we will show that each complementary region
of the geodesic lamination $\widetilde{\mathcal{L}}:=\mathrm{supp}(\mu)$ leads
to a well-defined barycenter. More precisely, given a complementary region
$\mathcal{R}$ of $\widetilde{\mathcal{L}}$, we will show in Proposition 5.5
that the map that to three points
$a,b,c\in(\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty))\cap X$
associates their barycenter is constant if the points are in different
connected components of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$. Here
$\mathcal{R}(\infty)$ is the intersection of $\partial{\mathcal{H}}^{2}$ with
the closure of $\mathcal{R}$ in $\overline{{\mathcal{H}}^{2}}$. In Lemma 5.7
we show that the obtained map is isometric.

As a first step in the proof of Proposition 5.5, we show that the crossratio
of 4-tuples separated by the lamination vanishes:

######  Lemma 5.4.

Let $\mathcal{R}$ be a complementary region of $\widetilde{\mathcal{L}}$ and
let $(a,b,c,d)$ be positively oriented such that
$\\{a,b,c,d\\}\subset\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$
and $\\{a,b\\}$, as well as $\\{c,d\\}$, are in different connected components
of $\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$. Then

$\mu(I_{[d,a]}\times I_{[b,c]})=0\,.$

In particular, if in addition $(a,b,c,d)\in X^{[4]}$

$[a,b,c,d]=0.$

###### Proof.

Since $\mathrm{supp}(\mu)=\widetilde{\mathcal{L}}$, we have that
$\mu((\partial{\mathcal{H}}^{2})^{(2)}\smallsetminus\widetilde{\mathcal{L}})=0$.
Thus it suffices to show that under the hypotheses of the lemma

$I_{[d,a]}\times
I_{[b,c]}\subset(\partial{\mathcal{H}}^{2})^{(2)}\smallsetminus\widetilde{\mathcal{L}}\,.$

Assume there is a geodesic $g\in\widetilde{\mathcal{L}}$ connecting
$I_{[d,a]}$ to $I_{[b,c]}$. Then $\mathcal{R}$ must be contained in one of the
half planes determined by $g$ and hence either $\\{a,b\\}$ or $\\{c,d\\}$ are
contained in the same connected component of
$(\partial{\mathcal{H}}^{2})^{(2)}\smallsetminus\mathcal{R}(\infty)$,
contradicting the hypothesis. The second statement follows directly from
Proposition 4.3 (1). ∎

With the use of Lemma 5.4 we proceed to define the barycenter of a
complementary region of $\widetilde{\mathcal{L}}$.

######  Proposition 5.5.

Let $\mathcal{R}$ be a complementary region of $\widetilde{\mathcal{L}}$. Then
$\beta(a,b,c)$ is independent of the choice of $\\{a,b,c\\}\subset X$,
provided $a,b,c$ lie in three distinct connected components of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$.

###### Proof.

We split the proof in three easy steps.

(1) Given three connected components $I_{1},I_{2},I_{3}$ of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$, $\beta(a,b,c)$
is independent of the choices $a\in I_{1}\cap X$, $b\in I_{2}\cap X$ and $c\in
I_{3}\cap X$.

Indeed, given $\\{a,a^{\prime}\\}\subset I_{1}\cap X$, we may assume, modulo
exchanging $b$ and $c$, and also $a$ and $a^{\prime}$, that
$(c,a,a^{\prime},b)\in X^{[4]}$. By Lemma 5.4 we have then that

$d_{X}(\beta(c,a,b),\beta(c,a^{\prime},b))=[c,a,a^{\prime},b]=0\,.$

Thus $\beta(a,b,c)=\beta(a^{\prime},b,c)$.

Given now $b^{\prime}\in I_{2}\cap X$ and $c^{\prime}\in I_{3}\cap X$, we
conclude, using the $S_{3}$-invariance of $\beta$, that

$\beta(a^{\prime},b^{\prime},c^{\prime})=\beta(a^{\prime},b^{\prime},c)=\beta(a^{\prime},b,c)=\beta(a,b,c).$

(2) Let $I_{1},I_{2},I_{3},I_{4}$ be distinct components of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$,
$\\{a,b,c,c^{\prime}\\}\subset X$ with $a\in I_{1}$, $b\in I_{2}$, $c\in
I_{3}$ and $c^{\prime}\in I_{4}$. Then $\beta(a,b,c)=\beta(a,b,c^{\prime})$.

We distinguish two cases.

(2.a) If $c,c^{\prime}$ are in the same connected component of
$\partial{\mathcal{H}}^{2}\smallsetminus\\{a,b\\}$, then possibly upon
permuting $a,b$ and $c,c^{\prime}$, we may assume that $(b,c,c^{\prime},a)\in
X^{[4]}$. By Lemma 5.4 this implies that

$d_{X}(\beta(b,c,a),\beta(b,c^{\prime},a))=[b,c,c^{\prime},a]=0\,.$

Thus $\beta(b,c,a)=\beta(b,c^{\prime},a)$.

(2.b) If instead $c,c^{\prime}$ are in distinct connected components of
$\partial{\mathcal{H}}^{2}\smallsetminus\\{a,b\\}$, we may assume, possibly
permuting $a,b$ and $c,c^{\prime}$, that $(a,c^{\prime},b,c)$ is positively
oriented. Using (2.a) in the second and fourth equality we obtain

$\beta(a,b,c^{\prime})=\beta(a,c^{\prime},b)=\beta(a,c^{\prime},c)=\beta(c,a,c^{\prime})=\beta(c,a,b)=\beta(a,b,c)\,,$

which shows the assertion.

(3) We finish now the proof of the proposition. Let
$\\{a,b,c,a^{\prime},b^{\prime},c^{\prime}\\}\subset X$ with $a\in I_{1}$,
$b\in I_{2}$, $c\in I_{3}$, $a^{\prime}\in I_{1}^{\prime}$, $b^{\prime}\in
I_{2}^{\prime}$ and $c\in I_{3}^{\prime}$, where $I_{1},I_{2},I_{3}$ and
$I_{1}^{\prime},I_{2}^{\prime},I_{3}^{\prime}$ are distinct connected
components of $\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$. We
can assume, up to reordering the indices that $I_{j}\neq I_{k}$ for $j\neq k$.
Then it follows from (1) and (2) that

$\beta(a,b,c)=\beta(a^{\prime},b,c)=\beta(a^{\prime},b^{\prime},c)=\beta(a^{\prime},b^{\prime},c^{\prime})\,.$

∎

######  Definition 5.6.

The _barycenter_ $\beta(\mathcal{R})$ of a complementary region $\mathcal{R}$
of $\widetilde{\mathcal{L}}$ is the point $\beta(a,b,c)$ for any choice
$\\{a,b,c\\}\subset X$ of points lying in distinct components of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}(\infty)$.

Taking into account the discussion in §2.3, the following lemma concludes the
proof of Theorem 5.3.

######  Lemma 5.7.

For the distance $d_{\mu}$ on the set $\mathcal{V}(\mu)$ of complementary
regions of $\widetilde{\mathcal{L}}$, we have

$d_{\mathcal{X}}(\beta(\mathcal{R}_{1}),\beta(\mathcal{R}_{2}))=d_{\mu}(\mathcal{R}_{1},\mathcal{R}_{2})$

for all $\mathcal{R}_{1},\mathcal{R}_{2}\in\mathcal{V}(\mu)$.

###### Proof.

Let $(x_{1},y_{1})$ be the endpoints of the geodesic in
$\partial\mathcal{R}_{1}$ separating $\mathcal{R}_{1}$ from $\mathcal{R}_{2}$
and $(x_{2},y_{2})$ the endpoints of the geodesic in $\partial\mathcal{R}_{2}$
separating $\mathcal{R}_{2}$ from $\mathcal{R}_{1}$, ordered so that
$(x_{1},y_{1},x_{2},y_{2})\in X^{[4]}$. Choose $a,b\in I_{(x_{1},y_{1})}$ in
different connected components of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}_{1}(\infty)$ and $c,d\in
I_{(x_{2},y_{2})}$ in different connected components of
$\partial{\mathcal{H}}^{2}\smallsetminus\mathcal{R}_{2}(\infty)$ in such a way
that $(a,b,c,d)\in X^{[4]}$.

$\mathcal{R}_{1}$$y_{1}$$x_{1}$$l_{1}$$l_{2}$$y_{2}$$x_{2}$$\mathcal{R}_{2}$$a$$b$$c$$d$$p_{1}$$p_{2}$

Then

$d_{\mathcal{X}}(\beta(\mathcal{R}_{1}),\beta(\mathcal{R}_{2}))=d_{\mathcal{X}}(\beta(a,b,d),\beta(a,c,d))=[a,b,c,d]\,.$

Since the geodesics bounding $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are all
$\mu$-short, we have

$\mu(\\{a\\}\times I_{[b,c]})\leq\mu(I_{(l_{1},l_{2})}\times
I_{(l_{2},l_{1})})=0\,,$

where $\\{l_{1},l_{2}\\}$ is the geodesic in $\partial\mathcal{R}_{1}$
separating $a$ from $\mathcal{R}_{1}$, and similarly $\mu(\\{d\\}\times
I_{[b,c]})=0$. As a result, from Proposition 4.3 (1) we have the equality

$\mu(I_{[d,a]}\times I_{[b,c]})=[a,b,c,d]=\mu(I_{(d,a)}\times I_{(b,c)})\,.$

If now $p_{i}\in\mathcal{R}_{i}$, the set of leaves in
$\widetilde{\mathcal{L}}$ that intersect the segment $(p_{1},p_{2})$ is
exactly the set of leaves in $\widetilde{\mathcal{L}}$ that separate
$\mathcal{R}_{1}$ from $\mathcal{R}_{2}$. This is also the same as the set of
leaves in $\widetilde{\mathcal{L}}$ that connect $I_{[d,a]}$ to $I_{[b,c]}$.
The assertion then follows from the above considerations, recalling that
$d_{\mu}(\mathcal{R}_{1},\mathcal{R}_{2})$ is the measure of this set,

$d_{\mathcal{X}}(\beta(\mathcal{R}_{1}),\beta(\mathcal{R}_{2}))=[a,b,c,d]=\mu(I_{[d,a]}\times
I_{[b,c]})=d_{\mu}(\mathcal{R}_{1},\mathcal{R}_{2})\,.$

∎

###  5.2. Framed actions on trees

Theorem 5.3 applies to framed actions on trees:

######  Proposition 5.8.

Let $\rho\colon\Gamma\to\mathrm{Isom}(\mathcal{T})$ be an action by isometries
on an real tree $\mathcal{T}$ with a framing $\varphi\colon
X\to\partial_{\infty}\mathcal{T}$. Suppose that the associated crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\varphi}$ is positive, and denote
by $\mu_{\rho}$ the associated geodesic current. Then $\mu_{\rho}$ corresponds
to a measured lamination, and there is a $\Gamma$-equivariant isometric
embedding

$\mathcal{T}(\mu_{\rho})\hookrightarrow\mathcal{T}$

In particular, for all hyperbolic $\gamma\in\Gamma$,

$\ell_{\mathcal{T}}(\rho(\gamma))=i(\mu_{\rho},\gamma).$

###### Proof.

By Proposition 3.8, the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\varphi}$ is ultrametric, hence by
Proposition 4.7, the current $\mu_{\rho}$ is of lamination type.

Now define the barycenter $\beta(x,y,z)$ of $(x,y,z)\in X^{(3)}$ as the
barycenter $\beta_{\mathcal{T}}(\varphi(x),\varphi(y),\varphi(z))$ in the tree
$\mathcal{T}$ of $(\varphi(x),\varphi(y),\varphi(z))$. Then $\beta$ is by
construction a equivariant barycenter map compatible with the crossratio (by
(4) in Example 3.5 and $S_{3}$-invariance of $\beta_{\mathcal{T}}$), hence
induces an equivariant isometry

$\Psi\colon\mathcal{V}(\mu_{\rho})\hookrightarrow\mathcal{T}$

of the vertices of the dual tree $\mathcal{T}(\mu_{\rho})$ by Theorem 5.3.
Since $\mathcal{T}$ is uniquely geodesic, we can extend $\Psi$ to
$\mathcal{T}(\mu_{\rho})$. Then for all $\gamma\in\Gamma$ we have
$\ell_{\mathcal{T}}(\rho(\gamma))=\ell_{\Psi(\mathcal{T}(\mu_{\rho}))}(\gamma)$
as $\Psi(\mathcal{T}(\mu_{\rho}))$ is a convex subset of $\mathcal{T}$, and
$\ell_{\Psi(\mathcal{T}(\mu_{\rho}))}(\gamma)=\ell_{\mathcal{T}(\mu_{\rho})}(\gamma)$
as $\Psi$ is isometric. Now for hyperbolic $\gamma$ representing a closed
geodesic $c$ we have
$\ell_{\mathcal{T}(\mu_{\rho})}(\gamma)=i(\mu_{\rho},\delta_{c})$, hence
$\ell_{\mathcal{T}}(\rho(\gamma))=i(\mu_{\rho},\gamma)$. ∎

##  6\. The geometry of the Siegel spaces over real closed fields

The goal of this section is to recall facts about the geometry of Siegel
spaces over real closed fields needed to show that maximal framed actions give
rise to a positive crossratio and admit a compatible barycenter (Lemma 6.6 and
Proposition 6.7).

###  6.1. Real closed fields

Recall that an ordered field is a field $\mathbf{F}$ endowed with a total
order relation $\leq$ satisfying:

  1. (1)

if $x\leq y$ then $x+z\leq y+z$ for all $z\in\mathbf{F}$;

  2. (2)

if $0\leq x$ and $0\leq y$, then $0\leq xy$.

The fields $\mathbf{Q}$ and $\mathbf{R}$ with their usual order are examples;
while some fields admit no ordering, like $\mathbf{C}$, others admit many,
like $\mathbf{R}(X)$.

######  Example 6.1.

The orders on $\mathbf{R}(X)$ admit the following description for
$\epsilon\in\\{-,+\\}$:

  1. (1)

$>_{\epsilon\infty}$: if $f\in\mathbf{R}(X)$, $f>_{\epsilon\infty}0$ if
$f(t)>0$ for $t\to\epsilon\infty$.

  2. (2)

$>_{a\epsilon}$ for $a\in\mathbf{R}$: we say that $f>_{a+}0$ if there exists
$\eta=\eta(f)\in\mathbf{R}$, $\eta>0$, with $f(t)>0$ on the interval
$(a,a+\eta)$, and $f>_{a-}0$ if $f(t)>0$ on the interval $(a-\eta,a)$.

A basic fact is that every ordered field $\mathbf{F}$ admits a real closure
$\overline{\mathbf{F}}^{\mathrm{r}}$, that is a maximal algebraic extension of
$\mathbf{F}$ to which the order extends. Such a real closure is then unique up
to a unique $\mathbf{F}$-isomorphism. An ordered field $\mathbf{F}$ is then
real closed if the ordering does not extend to any proper algebraic extension.
Two useful characterizations are the following:

  1. (1)

the field $\mathbf{F}$ is ordered and $\mathbf{F}(\imath)$ is algebraically
closed, with $\imath=\sqrt{-1}$;

  2. (2)

the field $\mathbf{F}$ is ordered, every positive element is a square and any
odd degree polynomial has a root.

Real closed fields have the same first order logic as the field $\mathbf{R}$
of the reals. An important consequence that is implicit in most of the
geometric properties of the Siegel space we use, is that any symmetric matrix
with coefficient in a real closed field is orthogonally similar to a diagonal
one.

An _$\mathbf{R}$ -valued valuation_ is a map

$v\colon\mathbf{F}\to\mathbf{R}\cup\\{\infty\\},$

where $v\colon(\mathbf{F}^{\times},\cdot)\to(\mathbf{R},+)$ is a group
homomorphism, $v(0)=\infty$, and, if we define the _norm_ of $x\in\mathbf{F}$
by $\|x\|_{v}:=e^{-v(x)}$ if $x\neq 0$ and $\|0\|_{v}:=0$, then

$\|x+y\|_{v}\leq\|x\|_{v}+\|y\|_{v}.$

An important valuation to keep in mind is
$-\ln(|\cdot|)\colon\mathbf{R}\to\mathbf{R}\cup\\{\infty\\}$. Indeed
valuations offer a replacement for the logarithm in more general real closed
fields. The valuation is order compatible if $|x|\leq|y|$ implies that
$v(x)\geq v(y)$, and it is non-Archimedean if $v(x+y)\geq\min\\{v(x),v(y)\\}$,
that is if $\|x+y\|_{v}\leq\max\\{\|x\|_{v},\|y\|_{v}\\}$.

######  Example 6.2.

The following are examples of ordered fields:

  1. (1)

The field $\mathbf{R}$ of real numbers and the field
$\overline{\mathbf{Q}}^{r}$ of real algebraic numbers; both are real closed.

  2. (2)

Let $\omega$ be a non-principal ultrafilter on $\mathbf{N}$. The quotient
$\mathbf{R}_{\omega}$ of the ring $\mathbf{R}^{\mathbf{N}}$ by the equivalence
relation $(x_{n})\sim(y_{n})$ if $\omega(\\{n:\,x_{n}=y_{n}\\})=1$, ordered in
such a way that positive elements are the classes of the sequences $(x_{n})$
such that $\omega(\\{n:\,x_{n}>0\\})=1$, is a real closed field called the
field of the hyperreals. It does not admit any order compatible
$\mathbf{R}$-valued valuation.

  3. (3)

Let $\sigma\in\mathbf{R}_{\omega}$ be a positive infinitesimal, that is
$\sigma$ can be represented by a sequence $(\sigma_{n})_{n\geq 0}$ with
$\lim\sigma_{n}=0$ and $\sigma_{n}>0$. Then

$\mathcal{O}_{\sigma}:=\left\\{x\in\mathbf{R}_{\omega}:\,|x|<\sigma^{-k}\text{
for some }k\in\mathbf{Z}\right\\}$

is a valuation ring with maximal ideal

$\mathcal{I}_{\sigma}=\left\\{x\in\mathbf{R}_{\omega}:\,|x|<\sigma^{k}\text{
for all }k\in\mathbf{Z}\right\\}\,.$

The quotient
$\mathbf{R}_{\omega,\sigma}:=\mathcal{O}_{\sigma}/\mathcal{I}_{\sigma}$ is a
real closed field, called the Robinson field. It admits an order compatible
valuation

$v(x)=\lim_{\omega}\frac{\ln|x_{n}|}{\ln\sigma_{n}}\,,$

where $(x_{n})_{n\geq 0}$ represents $x$, that leads to a non-Archimedean norm

$\|x\|_{v}:=e^{-v(x)}\,.$

  4. (4)

Let $G$ be a totally ordered Abelian group and let

$\mathcal{H}(G):=\left\\{f=\sum_{\sigma\in
G}a_{\sigma}X^{\sigma}:\,a_{\sigma}\in\mathbf{R}\text{ and
}\mathrm{supp}(f)\subset G\text{ is well ordered}\right\\}$

be the set of formal power series with exponents in $G$ and coefficients in
$\mathbf{R}$, where $\mathrm{supp}(f):=\\{\sigma\in G:\,a_{\sigma}\neq 0\\}$.
This is an $\mathbf{R}$-vector space and the restriction on supports allows
one to define a ring structure that extends the ordinary multiplication on the
group ring $\mathbf{R}[G]$ of $G$. In fact $\mathcal{H}(G)$ turns out to be a
field, called the Hahn field with exponents $G$. It is ordered by setting
$f>0$ if $a_{\sigma_{0}}>0$, where $\sigma_{0}=\min(\mathrm{supp}(f))$ and it
admits a $G$-valued compatible valuation $v(f)=\sigma_{0}$. Moreover
$\mathcal{H}(G)$ is real closed if $G$ is divisible. For all of the above
statements, see [DW96, Theorem 2.15].

  5. (5)

(Compare with § 8.3) If $\alpha\in\mathbf{R}\smallsetminus\mathbf{Q}$, then
the ring morphism

$\displaystyle\mathbf{R}[x,y]$
$\displaystyle\longrightarrow\,\,\,\mathcal{H}(\mathbf{R})$ $\displaystyle P$
$\displaystyle\longmapsto P(x,x^{\alpha})$

extends to $\mathbf{R}(x,y)$. In this way we obtain an order on
$\mathbf{R}(x,y)$ for which $P\in\mathbf{R}(x,y)$ is positive if and only if
for some $\epsilon>0$, $P(t,t^{\alpha})>0$ for all $t\in(0,\epsilon).$

###  6.2. The Siegel upper half space and the space
$\mathcal{B}_{n}^{\mathbf{F}}$

Let $\mathbf{F}$ be a real closed field, and $\imath$ be a square root of
$-1$. Endow $V=\mathbf{F}^{2n}$ with the standard symplectic form

$\left\langle\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix},\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}\right\rangle:={}^{t}x_{1}y_{2}-{}^{t}y_{1}x_{2}\,,$

where $x_{i},y_{i}\in\mathbf{F}^{n}$. The vector space
$\mathrm{Sym}_{n}(\mathbf{F})$ of symmetric matrices admits a partial order
defined by setting

$X\ll Y\quad\text{ if }Y-X\text{ is positive definite.}$

The Siegel upper half space is the semialgebraic set

$\mathcal{S}^{n}_{\mathbf{F}}:=\\{Z=X+\imath
Y:\,X,Y\in\mathrm{Sym}_{n}(\mathbf{F})\text{ and }Y\gg 0\\}$

on which

$\operatorname{Sp}(2n,\mathbf{F}):=\left\\{g=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}:\,\begin{array}[]{l}{{}^{t}\\!A}D-{}^{t}CB=\mathrm{Id},\\\
{}^{t}\\!AC={}^{t}CA,\\\ {}^{t}BD={}^{t}DB\end{array}\right\\}$

acts by fractional linear transformations

(8) $g_{\ast}Z:=(AZ+B)(CZ+D)^{-1}\,,$

transitively. Of course this action descends to an action of
$\operatorname{PSp}(2n,\mathbf{F})$. The stabilizer of
$\imath\mathrm{Id}_{n}\in\mathcal{S}^{n}_{\mathbf{F}}$ in
$\operatorname{Sp}(2n,\mathbf{F})$ is

$K=\operatorname{Sp}(2n,\mathbf{F})\cap\operatorname{O}(2n)=\left\\{\begin{pmatrix}\hphantom{-}A&B\\\
-B&A\end{pmatrix}:\,{}^{t}AA+{}^{t}BB=\mathrm{Id}_{n},\,{}^{t}AB\text{ is
symmetric}\right\\}\,.$

If $\mathbf{F}=\mathbf{R}$, then $\mathcal{S}^{n}_{\mathbf{R}}$ is the
symmetric space associated to $\operatorname{PSp}(2n,\mathbf{R})$. If,
instead, the real closed field $\mathbf{F}$ is endowed with an order
compatible non-Archimedean valuation $v$, then
$\operatorname{PSp}(2n,\mathbf{F})$ acts by isometries on a
$v(\mathbf{F})$-metric space $\mathcal{B}_{n}^{\mathbf{F}}$: a metric quotient
of $\mathcal{S}^{n}_{\mathbf{F}}$ whose construction we now recall. See
[BIPP21c, BIPP21a] for generalizations of this construction.

On $\mathcal{S}^{n}_{\mathbf{F}}$ we define an multiplicative
$\mathbf{F}$-valued distance function as follows. Since $\mathbf{F}$ is real
closed, any pair $(Z_{1},Z_{2})$ with $Z_{i}\in\mathcal{S}^{n}_{\mathbf{F}}$,
for $i=1,2$, is $\operatorname{PSp}(2n,\mathbf{F})$-congruent to a unique pair
$(\imath\,\mathrm{Id}_{n},\imath D)$, where
$D=\mathrm{diag}(d_{1},\dots,d_{n})$, $d_{1}\geq\cdots\geq d_{n}\geq 1$ in
$\mathbf{F}$. We then set

$D(Z_{1},Z_{2}):=\prod_{i=1}^{n}d_{i}.$

######  Proposition 6.3.

$D$ is a $\operatorname{PSp}(2n,\mathbf{F})$-invariant multiplicative distance
function on $\mathcal{S}^{n}_{\mathbf{F}}$, namely, for all
$Z_{1},Z_{2},Z_{3}\in\mathcal{S}^{n}_{\mathbf{F}}$,

(MD1):

$D(Z_{1},Z_{2})\in\mathbf{F}_{\geq 1}$, with equality of and only if
$Z_{1}=Z_{2}$ ;

(MD2):

$D(Z_{1},Z_{2})=D(Z_{2},Z_{1})$ ;

(MD3):

$D(Z_{1},Z_{2})\leq D(Z_{1},Z_{3})D(Z_{3},Z_{2})$ .

###### Proof.

(MD1) and (MD2) are clear. We consider the standard action of
$\operatorname{Sp}(2n,\mathbf{F})$ on $W=\wedge^{n}(\mathbf{F}^{2n})$, endowed
with the standard scalar product, which is $K$-invariant as
$K\subset\operatorname{O}(2n)$. For
$a=\mathrm{diag}(a_{1},\dots,a_{n},a_{1}^{-1},\dots,a_{n}^{-1})$ with
$a_{1}\geq\cdots\geq a_{n}\geq 1$ in $\mathbf{F}$, we easily see that

$\max_{w\in W,w\neq 0}\frac{||aw||}{||w||}=\prod_{i=1}^{n}a_{i}$

(the biggest eigenvalue of $a$ in $W$). For
$g\in\operatorname{Sp}(2n,\mathbf{F})$ the operator norm of $g$ on $W$ is
given by

$|||g|||:=\max_{w\in W,w\neq 0}\frac{||gw||}{||w||}\,.$

Since $g=kak^{\prime}$ for some $k,k^{\prime}$ in $K$ and $a$ as before (by
the Cartan decomposition), and

$D(\imath\mathrm{Id}_{n},g_{\ast}\imath\mathrm{Id}_{n})=D(\imath\mathrm{Id}_{n},a_{\ast}\imath\mathrm{Id}_{n})=2|||a|||=2|||g|||\,,$

(MD3) follows from submultiplicativity of the operator norm and transitivity
of the action of $\operatorname{Sp}(2n,\mathbf{F})$. ∎

On $\mathcal{S}^{n}_{\mathbf{F}}$ we define an _associated pseudo-distance_
$d^{1}$ as follows.

(9) $d^{1}(Z_{1},Z_{2}):=-v(D(Z_{1},Z_{2})=-\sum_{i=1}^{n}v(d_{i}).$

The triangle inequality for $d^{1}$ comes from (MD3). We denote by
${\mathcal{B}_{n}^{\mathbf{F}}}$ the metric quotient of
$\mathcal{S}^{n}_{\mathbf{F}}$ with respect to the pseudo-distance $d^{1}$. We
will not need it, but note in passing that ${\mathcal{B}_{n}^{\mathbf{F}}}$
can be identified with the quotient
$\operatorname{PSp}(2n,\mathbf{F})/\operatorname{PSp}(2n,\mathbf{U})$, where
$\mathbf{U}:=\\{x\in\mathbf{F}:\,\|x\|_{v}\leq 1\\}$.

###  6.3. Embedding in $\mathbf{K}$-Lagrangians

In the classical case, the Borel embedding of $\mathcal{S}^{n}_{\mathbf{R}}$
into the complex Grassmannian provides a way to endow
$\mathcal{S}^{n}_{\mathbf{R}}$ with structures defined on the Grassmannian,
such as, for example, a crossratio. We recall from [BP17] the analogous
picture in the case of a general real closed field.

Let $\mathbf{K}=\mathbf{F}(\imath)$ be the algebraic closure of $\mathbf{F}$
and let us also denote by $\langle\,\,,\,\,\rangle$ the $\mathbf{K}$-linear
extension of the standard symplectic form to $\mathbf{K}^{2n}$ and by
$\sigma\colon\mathbf{K}^{2n}\to\mathbf{K}^{2n}$ the complex conjugation. Given
matrices $Z_{1},Z_{2}\in M_{n}(\mathbf{K})$, we will denote by
$\left\langle\begin{pmatrix}Z_{1}\\\ Z_{2}\end{pmatrix}\right\rangle$ the
subspace of $\mathbf{K}^{2n}$ generated by the column vectors. We denote by
$\mathcal{L}(\mathbf{K}^{2n})$ the submanifold of ${\rm
Gr}_{n}(\mathbf{K}^{2n})$ consisting of subspaces that are isotropic for the
form $\langle\,\,,\,\,\rangle$. The map

$\displaystyle\mathrm{Sym}_{n}(\mathbf{K})$
$\displaystyle\longrightarrow\,\,\,\mathcal{L}(\mathbf{K}^{2n})$
$\displaystyle Z$ $\displaystyle\longmapsto\left\langle\begin{pmatrix}Z\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$

gives a bijection between $\mathrm{Sym}_{n}(\mathbf{K})$ and the subset of
$\mathcal{L}(\mathbf{K}^{2n})$ of all Lagrangians transverse to
$\ell_{\infty}:=\left\langle\begin{pmatrix}\mathrm{Id}_{n}\\\
0\end{pmatrix}\right\rangle$. This map intertwines the action of
$\operatorname{PSp}(2n,\mathbf{K})$ on $\mathrm{Sym}_{n}(\mathbf{K})$ by
fractional linear transformations (8) and the standard action on
$\mathcal{L}(\mathbf{K}^{2n})$. This bijection maps
$\mathcal{S}^{n}_{\mathbf{F}}$ to the projective model

$\mathcal{D}_{\mathbf{F}}:=\\{L\in\mathcal{L}(\mathbf{K}^{2n}):\,-\imath\langle\,\cdot\,,\sigma(\,\cdot\,)\rangle|_{L\times
L}\gg 0\\}$

and sends $\mathrm{Sym}_{n}(\mathbf{F})$ to

$\\{\ell\otimes\mathbf{K}:\,\ell\in\mathcal{L}(\mathbf{F}^{2n}),\,\ell\text{
is transverse to }\ell_{\infty}\\}\,.$

###  6.4. Maximal triples and intervals

We associate to a triple $(\ell_{1},\ell_{2},\ell_{3})$ of pairwise transverse
Lagrangians in $\mathcal{L}(\mathbf{F}^{2n})$ the quadratic form
$Q_{(\ell_{1},\ell_{2},\ell_{3})}$ on $\ell_{1}$ defined by

$Q_{(\ell_{1},\ell_{2},\ell_{3})}(v):=\langle v,v^{\prime}\rangle\,,$

where $v^{\prime}\in\ell_{3}$ is the unique vector such that
$v+v^{\prime}\in\ell_{2}$. If $\ell=\left\langle\begin{pmatrix}X\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$ and
$\ell^{\prime}=\left\langle\begin{pmatrix}X^{\prime}\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$ are pairwise transverse, then in
the coordinates

(10) $\displaystyle\mathbf{F}^{n}$ $\displaystyle\longrightarrow\ell$
$\displaystyle w$ $\displaystyle\mapsto\begin{pmatrix}X\\\
\mathrm{Id}_{n}\end{pmatrix}w$

the quadratic form $Q_{(\ell,\ell^{\prime},\ell_{\infty})}$ is represented by
$X^{\prime}-X$.

Two triples of pairwise transverse Lagrangians $(\ell_{1},\ell_{2},\ell_{3})$
and $(m_{1},m_{2},m_{3})$ are $\operatorname{PSp}(2n,\mathbf{F})$-congruent if
and only if the quadratic spaces $(\ell_{1},Q_{(\ell_{1},\ell_{2},\ell_{3})})$
and $(m_{1},Q_{(m_{1},m_{2},m_{3})})$ are isomorphic or equivalently if and
only if $Q_{(\ell_{1},\ell_{2},\ell_{3})}$ and $Q_{(m_{1},m_{2},m_{3})}$ have
the same signature [BP17, Proposition 2.5]. The value of the Maslov cocycle on
$(\ell_{1},\ell_{2},\ell_{3})$ is the signature of
$Q_{(\ell_{1},\ell_{2},\ell_{3})}$

$\tau(\ell_{1},\ell_{2},\ell_{3}):=\operatorname{sign}Q_{(\ell_{1},\ell_{2},\ell_{3})}\,.$

The triple $(\ell_{1},\ell_{2},\ell_{3})$ is maximal if
$\tau(\ell_{1},\ell_{2},\ell_{3})=n$, the maximal value the Maslov cocycle can
take. Similarly we say that a triple $(\ell_{1},\ell_{2},\ell_{3})$ is minimal
if $\tau(\ell_{1},\ell_{2},\ell_{3})=-n$; it is easy to verify that
$(\ell_{1},\ell_{2},\ell_{3})$ is maximal if and only if
$(\ell_{2},\ell_{1},\ell_{3})$ is minimal. The group
$\operatorname{PSp}(2n,\mathbf{F})$ acts transitively on pairs of transverse
Lagrangians in $\mathcal{L}(\mathbf{F}^{2n})$ and on maximal triples.

Given $\ell,\ell^{\prime}\in\mathcal{L}(\mathbf{F}^{2n})$, we define the
_interval_

$I_{(\ell,\ell^{\prime})}:=\\{m\in\mathcal{L}(\mathbf{F}^{2n}):\,(\ell,m,\ell^{\prime})\text{
is maximal}\\}\,.$

######  Lemma 6.4 ([BP17, Lemma 2.10]).

Let $\ell=\left\langle\begin{pmatrix}X\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$ and
$\ell^{\prime}=\left\langle\begin{pmatrix}X^{\prime}\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$. If the triple
$(\ell,\ell^{\prime},\ell_{\infty})$ is maximal, then

$I_{(\ell,\ell^{\prime})}=\left\\{\left\langle\begin{pmatrix}Y\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle:\,X\ll Y\ll X^{\prime}\right\\}\,.$

For $X,X^{\prime}\in\mathrm{Sym}_{n}(\mathbf{F})$ with $X\ll X^{\prime}$, we
will also denote by $I_{(X,X^{\prime})}$ the set

$I_{(X,X^{\prime})}:=\\{Y\in\mathrm{Sym}_{n}(\mathbf{F}):\,X\ll Y\ll
X^{\prime}\\}$

and set

$I_{(X,\infty)}:=\\{Y\in\mathrm{Sym}_{n}(\mathbf{F}):\,X\ll Y\\}\,.$

###  6.5. Crossratios

In this subsection we recall the endomorphism valued crossratio from [BP17, §
4.1] on quadruples of Lagrangians. This, together with a maximal framing, will
allow us in § 7.1 to associate to any maximal framed representation $\rho$ a
positive crossratio as in § 3.

Given a quadruple of Lagrangians $(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$ in
$\mathcal{L}(\mathbf{F}^{2n})$ with
$\ell_{1}\cap\ell_{2}=\ell_{3}\cap\ell_{4}=\\{0\\}$, their _crossratio_ is the
endomorphism of $\ell_{1}$ given by

$R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})={\rm
p}_{\ell_{1}}^{\|\ell_{2}}\circ{\rm p}_{\ell_{4}}^{\|\ell_{3}}|_{\ell_{1}}\,,$

where ${\rm p}_{\ell_{j}}^{\|\ell_{i}}$ denotes the projection of
$\mathbf{F}^{2n}$ to $\ell_{j}$ parallel to the complementary space
$\ell_{i}$. One verifies that for all $g\in\operatorname{Sp}(2n,\mathbf{F})$

$R(g\ell_{1},g\ell_{2},g\ell_{3},g\ell_{4})=gR(\ell_{1},\ell_{2},\ell_{3},\ell_{4})g^{-1}$

and hence

$\det R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$

is $\operatorname{PSp}(2n,\mathbf{F})$-invariant. If, for $j\in\\{1,2,3,4\\}$,
$\ell_{j}\cap\ell_{\infty}=\\{0\\}$, then
$\ell_{j}=\left\langle\begin{pmatrix}X_{j}\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$ for some
$X_{j}\in\mathrm{Sym}_{n}(\mathbf{F})$. By [BP17, Lemma 4.2], the matrix
representing $R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$ in the basis of
$\ell_{1}$ given by (10) is

(11) $(X_{1}-X_{2})^{-1}(X_{2}-X_{4})(X_{3}-X_{4})^{-1}(X_{1}-X_{3})\,.$

We will denote such matrix $R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$, with an
abuse of notation.

######  Proposition 6.5 ([BP17, Lemma 4.4]).

Let $\ell_{1},\ell_{2},\ell_{3},\ell_{4},\ell_{5}$ be pairwise transverse
Lagrangians. Then:

  1. (1)

$R(\ell_{1},\ell_{2},\ell_{4},\ell_{5})=R(\ell_{1},\ell_{2},\ell_{3},\ell_{5})R(\ell_{1},\ell_{3},\ell_{4},\ell_{5})\,.$

  2. (2)

$R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})\text{ is conjugate to
}R(\ell_{3},\ell_{4},\ell_{1},\ell_{2})\,.$

  3. (3)

$\det R(\ell_{1},\ell_{2},\ell_{4},\ell_{5})=\det
R(\ell_{1},\ell_{2},\ell_{3},\ell_{5})\det
R(\ell_{1},\ell_{3},\ell_{4},\ell_{5})$ and
$\det R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})=\det
R(\ell_{3},\ell_{4},\ell_{1},\ell_{2})\,.$

  4. (4)

If $(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$ is a maximal quadruple, then all
eigenvalues of $R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$ belong to $\mathbf{F}$
and are strictly larger than one. In particular $\det
R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})>1$.

###  6.6. $\mathbf{F}$-tubes and orthogonal projections

If $\ell,\ell^{\prime}\in\mathcal{L}(\mathbf{F}^{2n})$, the _$\mathbf{F}$
-tube_ determined by $\ell,\ell^{\prime}$ is the semi-algebraic subset of
$\mathcal{S}^{n}_{\mathbf{F}}$ given by the equation

$\mathcal{Y}_{\ell,\ell^{\prime}}:=\left\\{Z\in\mathcal{S}^{n}_{\mathbf{F}}:\,R\left(\ell,\left\langle\begin{pmatrix}Z\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle,\left\langle\begin{pmatrix}\overline{Z}\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle,\ell^{\prime}\right)=-\mathrm{Id}_{n}\right\\}\,,$

where $\overline{Z}$ is the complex conjugate of $Z$, [BP17, § 4.2].

If $\mathbf{F}=\mathbf{R}$ and $n=1$, the $\mathbf{R}$-tube
$\mathcal{Y}_{\ell,\ell^{\prime}}$ is the geodesic between $\ell$ and
$\ell^{\prime}$ while, for $n\geq 1$, $\mathcal{Y}_{\ell,\ell^{\prime}}$ is a
symmetric subspace of $\mathcal{S}^{n}_{\mathbf{R}}$ that is a Lagrangian
submanifold and is isometric to the symmetric space associated to
$\operatorname{GL}(n,\mathbf{R})$. In general, for all
$g\in\operatorname{PSp}(2n,\mathbf{F})$,

(12) $\mathcal{Y}_{g\ell,g\ell^{\prime}}=g(\mathcal{Y}_{\ell,\ell^{\prime}})$

and if we denote

$\ell_{0}:=\left\langle\begin{pmatrix}0\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle\,,$

then

$\mathcal{Y}_{\ell_{0},\ell_{\infty}}:=\\{\imath
Y:\,Y\in\mathrm{Sym}_{n}(\mathbf{F}),Y\gg 0\\}\,.$

We will often write $\mathcal{Y}_{0,\infty}$ for
$\mathcal{Y}_{\ell_{0},\ell_{\infty}}$.

If $(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$ is a maximal quadruple, then
$\mathcal{Y}_{\ell_{1},\ell_{3}}$ and $\mathcal{Y}_{\ell_{2},\ell_{4}}$ meet
exactly in one point. Such $\mathbf{F}$-tubes are called orthogonal if

$R(\ell_{1},\ell_{2},\ell_{3},\ell_{4})=2\,\mathrm{Id}_{n}$

(see [BP17, Proposition 4.7 and Definition 4.14]). If $\mathbf{F}=\mathbf{R}$,
the tubes are orthogonal if and only if they are orthogonal as submanifolds of
the Riemannian manifold $\mathcal{S}^{n}_{\mathbf{R}}$. Given any point $p\in
I_{(\ell_{1},\ell_{3})}\cup I_{(\ell_{3},\ell_{1})}$, there exists a unique
$\mathbf{F}$-tube $\mathcal{Y}_{\ell_{2},\ell_{4}}$ orthogonal to
$\mathcal{Y}_{\ell_{1},\ell_{3}}$ “with endpoint $p$” in the following sense

$\begin{cases}\ell_{2}:=p\quad&\text{ if }p\in I_{(\ell_{1},\ell_{3})}\\\
\ell_{4}:=p\quad&\text{ if }p\in I_{(\ell_{3},\ell_{1})}\end{cases}$

In this way we obtain a map

${\rm pr}_{\mathcal{Y}_{\ell_{1},\ell_{3}}}:I_{(\ell_{1},\ell_{3})}\cup
I_{(\ell_{3},\ell_{1})}\to\mathcal{Y}_{\ell_{1},\ell_{3}}$

called the orthogonal projection.

In the case of $\mathcal{Y}_{0,\infty}$ the map ${\rm
pr}_{\mathcal{Y}_{0,\infty}}$ is given by

${\rm pr}_{\mathcal{Y}_{0,\infty}}(Y)=\begin{cases}\hphantom{-}\imath Y&\text{
if }Y\in I_{(\ell_{0},\ell_{\infty})}\\\ -\imath Y&\text{ if }Y\in
I_{(\ell_{\infty},\ell_{0})}\,.\end{cases}$

In view of (12), this implies in particular that the restrictions of ${\rm
pr}_{\mathcal{Y}_{\ell_{1},\ell_{3}}}$ to $I_{(\ell_{1},\ell_{3})}$ and
$I_{(\ell_{3},\ell_{1})}$ are both bijective.

######  Lemma 6.6.

Assume $(\ell,\ell_{1},\ell_{2},\ell^{\prime})$ is maximal. Then for the
pseudodistance $d^{1}$ on $\mathcal{S}^{n}_{\mathbf{F}}$ (see (9) in §6.2) we
have,

$d^{1}\left({\rm pr}_{\mathcal{Y}_{\ell,\ell^{\prime}}}(\ell_{1}),{\rm
pr}_{\mathcal{Y}_{\ell,\ell^{\prime}}}(\ell_{2})\right)=-v(\det
R(\ell,\ell_{1},\ell_{2},\ell^{\prime}))\,.$

###### Proof.

We may assume that $\ell=\ell_{0}$ and $\ell^{\prime}=\ell_{\infty}$, and set
$\ell_{j}:=\left\langle\begin{pmatrix}Y_{j}\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$, for $j=1,2$. Then we have $0\ll
Y_{1}\ll Y_{2}$ and in particular the eigenvalues $r_{1},\dots,r_{n}$ of the
symmetric matrix $Y_{1}^{-1/2}Y_{2}Y_{1}^{-1/2}$ are all greater than 1. By
invariance of the distance $d^{1}$ we have

$\displaystyle d^{1}(\imath Y_{1},\imath Y_{2})=$ $\displaystyle
d^{1}\left(\imath\mathrm{Id}_{n},\imath
Y_{1}^{-1/2}Y_{2}Y_{1}^{-1/2}\right)=\sum_{j=1}^{n}-v(r_{j})$ $\displaystyle=$
$\displaystyle-v\left(\prod_{j=1}^{n}r_{j}\right)=-v\left(\det(Y_{1}^{-1}Y_{2})\right)$
$\displaystyle=$ $\displaystyle-v(\det
R(\ell,\ell_{1},\ell_{2},\ell^{\prime}))\,,$

where the last equality uses (11). This proves the lemma. ∎

###  6.7. Barycenters

Assume that $\mathbf{F}$ is non-Archimedean. If the triple of Lagrangians
$(\ell_{1},\ell_{2},\ell_{3})$ is either maximal or minimal, then $\ell_{2}\in
I_{(\ell_{1},\ell_{3})}\cup I_{(\ell_{3},\ell_{1})}$. We define

$b(\ell_{1},\ell_{2},\ell_{3}):={\rm
pr}_{{\mathcal{Y}}_{\ell_{1},\ell_{3}}}(\ell_{2})\in\mathcal{S}^{n}_{\mathbf{F}}.$

The _barycenter_ of the triple $(\ell_{1},\ell_{2},\ell_{3})$ is the point

$B(\ell_{1},\ell_{2},\ell_{3})=\pi(b(\ell_{1},\ell_{2},\ell_{3}))\,,$

where $\pi\colon\mathcal{S}^{n}_{\mathbf{F}}\to\mathcal{B}_{n}^{\mathbf{F}}$
is the metric quotient introduced in §6.2. Our goal is to show the following
(cfr. [BP17, Lemma 7.5]):

######  Proposition 6.7.

$B(\ell_{1},\ell_{2},\ell_{3})$ is invariant under permutation of the
arguments.

First we need to establish a formula for the (pseudo-)distance in
$\mathcal{S}_{1}^{\mathbf{F}}$:

######  Lemma 6.8.

If $z_{1}=x_{1}+\imath y_{1},\;z_{2}=x_{2}+\imath
y_{2}\in\mathcal{S}_{1}^{\mathbf{F}}$, then

$d(z_{1},z_{2})=\max\left\\{-v\left(\frac{(x_{1}-x_{2})^{2}}{y_{1}y_{2}}\right),-v\left(\frac{y_{1}}{y_{2}}\right),-v\left(\frac{y_{2}}{y_{1}}\right)\right\\}\,,$

###### Proof.

Recall that for a general real closed field $\mathbf{F}$, if
$z_{1},z_{2}\in\mathcal{S}_{1}^{\mathbf{F}}$,

$d(z_{1},z_{2})=\ln\|T+\sqrt{T^{2}-1}\|_{v}\,,$

where $T=1+\frac{|z_{1}-z_{2}|^{2}}{2y_{1}y_{2}}$. Since $\mathbf{F}$ is non-
Archimedean, then for $a,b\in\mathbf{F}$, $a\geq 0$, $b\geq 0$, we have

$\|a+b\|_{v}=\|\max\\{a,b\\}\|_{v}\,.$

Hence

$\|T+\sqrt{T^{2}-1}\|_{v}=\|T\|_{v}$

and

$d(z_{1},z_{2})=\ln\|T\|_{v}\,.$

Since

$T=1+\frac{|z_{1}-z_{2}|^{2}}{2y_{1}y_{2}}=\frac{(x_{1}-x_{2})^{2}}{2y_{1}y_{2}}+\frac{y_{1}}{2y_{2}}+\frac{y_{2}}{2y_{1}}\,,$

then

$\|T\|_{v}=\max\left\\{\left\|\frac{(x_{1}-x_{2})^{2}}{y_{1}y_{2}}\right\|_{v},\left\|\frac{y_{1}}{y_{2}}\right\|_{v},\left\|\frac{y_{2}}{y_{1}}\right\|_{v}\right\\}\,,$

where we took into account that $\|n\|_{v}=1$ for any
$n\in\mathbf{Z}\setminus\\{0\\}$. ∎

###### Proof of Proposition 6.7.

Since $\operatorname{PSp}(2n,\mathbf{F})$ acts transitively on maximal and
minimal triples, we may assume that
$(\ell_{1},\ell_{2},\ell_{3}):=\left(\ell_{0},\left\langle\begin{pmatrix}Y\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle,\ell_{\infty}\right)$, where
$Y=\pm\mathrm{Id}_{n}$, depending on whether $(\ell_{1},\ell_{2},\ell_{3})$ is
maximal or minimal. A computation then gives:

$\displaystyle{\rm pr}_{\mathcal{Y}_{0,\infty}}(\pm\mathrm{Id}_{n})$
$\displaystyle=\imath\mathrm{Id}_{n}$ $\displaystyle{\rm
pr}_{\mathcal{Y}_{\ell_{\infty},\ell_{2}}}(0)$ $\displaystyle=(\pm
1+\imath)\mathrm{Id}_{n}$ $\displaystyle{\rm
pr}_{\mathcal{Y}_{\ell_{2},\ell_{0}}}(\infty)$ $\displaystyle=\frac{\pm
1+\imath}{2}\mathrm{Id}_{n}\,.$

Observe that if
$Z=\mathrm{diag}(z_{1},\dots,z_{n}),W=\mathrm{diag}(w_{1},\dots,w_{n})\in\mathcal{S}^{n}_{\mathbf{F}}$,
then

$d^{1}(Z,W)=\sum_{i=1}^{n}d(z_{i},w_{i})\,.$

Thus in order to compute the distances between various projections we just
have to compute the following in $\mathcal{S}_{1}^{\mathbf{F}}$:

$\displaystyle d(\imath,\imath\pm 1)$
$\displaystyle=\ln\max\\{\|1\|_{v},\|1\|_{v},\|1\|_{v}\\}=0$ $\displaystyle
d\left(\imath,\left(\frac{1+\imath}{2}\right)\right)$
$\displaystyle=\ln\max\left\\{\left\|\frac{1}{2}\right\|_{v},\left\|\frac{1}{2}\right\|_{v},\|2\|_{v}\right\\}=0\,.$

This concludes the proof. ∎

##  7\. Applications to maximal framed representations

In this section we prove Theorem 1.2, Corollary 1.3, Corollary 1.8, and
Theorem 1.5 in the introduction.

###  7.1. The geodesic current $\mu_{\rho}$ and Theorem 1.2

Let $\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$ be a maximal framed
representation with maximal framing
$\varphi\colon{H_{\Gamma}}\to\mathcal{L}(\mathbf{F}^{2n})$. Define then for
every $(x_{1},x_{2},x_{3},x_{4})\in{H_{\Gamma}}^{[4]}$

$[x_{1},x_{2},x_{3},x_{4}]_{\rho}:=-v(\det
R(\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3}),\varphi(x_{4})))\,.$

It follows from Proposition 6.5 that
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\rho}$ is a positive crossratio on
${H_{\Gamma}}$, and hence (Proposition 4.3 and 4.8) there is a geodesic
current $\mu_{\rho}$ such that

$\mathrm{per}(\gamma)=i(\mu_{\rho},\delta_{c})$

for every closed geodesic $c$ represented by a hyperbolic element
$\gamma\in\Gamma$.

Next, for the computation of $\mathrm{per}(\gamma)$, recall that
$g\in\operatorname{Sp}(2n,\mathbf{F})$ is called Shilov hyperbolic if there is
a $g$-invariant decomposition $\mathbf{F}^{2n}=\ell_{+}\oplus\ell_{-}$ into
Lagrangians such that all eigenvalues of $g|_{\ell_{-}}$ have absolute value
strictly smaller than one (and thus all eigenvalues of $g|_{\ell_{+}}$ have
absolute value strictly larger than one), [BP17, Definition 2.12]. An element
$g\in\operatorname{PSp}(2n,\mathbf{F})$ is Shilov hyperbolic if any of its
lifts to $\operatorname{Sp}(2n,\mathbf{F})$ is. Of course the decomposition is
uniquely determined by $g$ and doesn’t depend on the lift. We will need the
following:

######  Lemma 7.1 ([BP17, Lemma 7.9]).

Let $g\in\operatorname{Sp}(2n,\mathbf{F})$ be Shilov hyperbolic and let
$\lambda_{1},\dots,\lambda_{n}$ be the eigenvalues of $g|_{\ell_{+}}$. Then
for any $\ell\in I_{(\ell_{-},\ell_{+})}$ we have

$\det
R(\ell_{-},\ell,g\ell,\ell_{+})=\left(\prod_{i=1}^{n}\lambda_{i}\right)^{2}\,.$

###### Proof.

We may assume by transitivity of the action of
$\operatorname{Sp}(2n,\mathbf{F})$ on pairs of transverse Lagrangians, that
$(\ell_{-},\ell_{+})=(\ell_{0},\ell_{\infty})$. Furthermore
$g=\begin{pmatrix}A&0\\\ 0&{}^{t}A^{-1}\end{pmatrix}$, where $A$ is the matrix
of $g|_{\ell_{+}}$. Then $\ell$ corresponds to a matrix
$X\in\mathrm{Sym}_{n}(\mathbf{F})$ with $X\gg 0$ and (11) gives

$R(\ell_{-},\ell,g\ell,\ell_{+})=X^{-1}AXA^{t}\,.$

This implies that

$\det R(\ell_{-},\ell,g\ell,\ell_{+})=(\det A)^{2},$

hence the lemma. ∎

Let $\gamma\in\Gamma$ be hyperbolic with attractive and repulsive fixed points
$\gamma_{+}$ and $\gamma_{-}$. It follows from [BP17, Theorem 1.9] that
$\rho(\gamma)$ is Shilov hyperbolic with corresponding decomposition
$\mathbf{F}^{2n}=\varphi(\gamma_{-})\oplus\varphi(\gamma_{+})$. In particular
the framing is uniquely determined by the representation $\rho$. Furthermore,
if $\lambda_{1}(\gamma),\dots,\lambda_{n}(\gamma)$ are the eigenvalues of
$\rho(\gamma)$ with $|\lambda_{1}(\gamma)|\geq\dots|\lambda_{n}(\gamma)|\geq
1$, then Lemma 7.1 implies that if $x\in I_{(\gamma_{-},\gamma_{+})}$,

$\displaystyle\mathrm{per}(\gamma)$ $\displaystyle=-v(\det
R(\varphi(\gamma_{-}),\varphi(x),\rho(\gamma)\varphi(x),\varphi(\gamma_{+})))$
$\displaystyle=2\sum_{i=1}^{n}-v(\lambda_{i}(\gamma))=L(\rho(\gamma))\,.$

If there is an element $\gamma\in\Gamma$ with
$-v(\mathrm{tr}(\rho(\gamma)))>0$, then necessarily $\rho(\gamma)$ has an
eigenvalue with the same property and thus $\mathrm{per}(\gamma)>0$. Viceversa
if $\mu_{\rho}$ is non-zero, there exists a proper closed rectangle
$R=I_{[a,b]}\times I_{[c,d]}$ with $\mu_{\rho}(R)>0$. Since $\Gamma$ is a
lattice, we find $\gamma\in\Gamma$ with $\gamma_{+}\in I_{(b,c)},\gamma_{-}\in
I_{(d,a)}$. For such $\gamma$, the intersection
$i(\mu_{\rho},\delta_{\gamma})>0$, and thus $\mathrm{per}(\gamma)>0$, which
implies that $\|\lambda_{1}(\gamma)\ldots\lambda_{n}(\gamma)\|_{v}>1$. Now
assume by contradiction that $-v(\mathrm{tr}(\rho(\gamma^{s})))\leq 0$ for all
$s\in\mathbf{N}$. Then the coefficients of the characteristic polynomial of
$\wedge^{n}\rho(\gamma)$ belong to the ring
$\mathcal{O}:=\\{x\in\mathbf{F}|\,\|x\|_{v}\leq 1\\}$ as well. But
$\lambda_{1}(\gamma)\ldots\lambda_{n}(\gamma)\in\mathbf{F}$ is a root of this
monic polynomial, and since $\mathcal{O}\subset\mathbf{F}$ is a valuation
ring, it is integrally closed in $\mathbf{F}$ (see [EP05, Theorem 3.1.3.(1)]).
This implies $\|\lambda_{1}(\gamma)\ldots\lambda_{n}(\gamma)\|_{v}\leq 1$, a
contradiction, that concludes the proof of Theorem 1.2.

###  7.2. Displacing representations and Corollary 1.3

Assume that $\Sigma$ is compact and let
$\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$ be a maximal framed
representation. If

$\mathrm{Syst}(\rho):=\inf_{\gamma\neq e}L(\rho(\gamma))>0\,,$

it follows from Theorem 1.2 that the associated geodesic current $\mu_{\rho}$
has $\mathrm{Syst}(\mu_{\rho})>0$; hence there exist $c_{1},c_{2}$ such that,
for every $\gamma\in\Gamma$,

(13) $c_{1}\ell(\gamma)\leq L(\rho(\gamma))\leq c_{2}\ell(\gamma)\,,$

where $\ell(\gamma)$ is the hyperbolic length of $\gamma$ [BIPP19, Theorem
1.3]. It follows then from [DGLM11] that the $\Gamma$-action on
$\mathcal{B}_{n}^{\mathbf{F}}$ induced by $\rho$ is displacing and hence, for
every $x\in\mathcal{B}_{n}^{\mathbf{F}}$, the map $\gamma\mapsto\rho(\gamma)x$
is a quasi-isometric embedding. This proves Corollary 1.3.

###  7.3. Maximal representations in $\operatorname{PSp}(2n,\mathbf{R})$ and
Corollary 1.8

If $\Sigma$ is not necessarily compact, the inequalities in (13) do not
necessarily hold. However, applying [BIPP19, Corollary 1.5 (2)] to the current
$\mu_{\rho}$, we deduce following

######  Corollary 7.2.

Let $\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{F})$ be a representation
admitting a maximal framing defined on ${H_{\Gamma}}$. Assume that
$\mathrm{Syst}_{\Sigma}(\rho)>0$. Then for every compact subset
$K\subset\Sigma$, there are constants $0<c_{1}\leq c_{2}$ such that

$c_{1}\ell(c)\leq L(\rho(\gamma))\leq c_{2}\ell(c)$

for every $\gamma\in\Gamma$ representing a closed geodesic $c$ contained in
$K$. In particular there exist constants $c_{1},c_{2}$ such that this holds
for all $\gamma$ representing simple closed geodesics.

Assume now that $\mathbf{F}=\mathbf{R}$ and let
$\rho\colon\Gamma\to\operatorname{PSp}(2n,\mathbf{R})$ be a maximal
representation. Then there is a maximal framing $\varphi$ defined on
$\partial{\mathcal{H}}^{2}$ [BIW10, Theorem 8] and hence, by Theorem 1.2, a
geodesic current $\mu_{\rho}$ with

$i(\mu_{\rho},\delta_{c})=L(\rho(\gamma))\,,$

for every closed geodesic $c$ represented by a hyperbolic element
$\gamma\in\Gamma$. The Collar Lemma, [BP17, Theorem 1.9], then implies that if
$\gamma,\eta$ are intersecting hyperbolic elements,

$\left(e^{\frac{L(\rho(\gamma))}{n}}-1\right)\left(e^{\frac{L(\rho(\eta))}{n}}-1\right)\geq
1\,.$

This implies that, if $\gamma$ is self-intersecting,

$L(\rho(\gamma))\geq n(\ln 2)\,,$

and that there are at most $3g-3+p$ conjugacy classes of hyperbolic elements
$\gamma$ with $L(\rho(\gamma))\leq n(\ln 2)$. In particular
$\mathrm{Syst}(\mu_{\rho})>0$. Corollary 1.8 then follows from Corollary 7.2.

###  7.4. Lamination type currents and Theorem 1.5

Let $\mathbf{F}$ be non-Archimedean. Given a maximal framed representation, as
above, and with the notation of § 6.7, define the _barycenter_ of
$(x,y,z)\in{H_{\Gamma}}^{(3)}$

$\beta_{\rho}(x,y,z):=B(\varphi(x),\varphi(y),\varphi(z))\in\mathcal{B}_{n}^{\mathbf{F}}\,.$

It follows from Proposition 6.7 that $\beta$ is indeed a barycenter according
to Definition 5.1 and from Lemma 6.6 that it is compatible with the crossratio
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\rho}$ defined in § 7.1.

If now $\mu_{\rho}$ is of lamination type, we deduce from Theorem 5.3 that
there is a well defined equivariant isometric embedding

$\mathcal{V}({\mu_{\rho}})\hookrightarrow\mathcal{B}_{n}^{\mathbf{F}}$

from the set of vertices $\mathcal{V}({\mu_{\rho}})$ of the $\mathbf{R}$-tree
$\mathcal{T}(\mu_{\rho})$ associated to $\mu_{\rho}$ into the metric space
$\mathcal{B}_{n}^{\mathbf{F}}$.

###  7.5. The value group of
$[\,\cdot\,,\,\cdot\,,\,\cdot\,,\,\cdot\,]_{\rho}$ and Theorem 1.4

Let $\rho\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$ be maximal framed
with framing $\varphi\colon{H_{\Gamma}}\to\mathcal{L}(\mathbf{F}^{2n})$, and
set as always

$[x_{1},x_{2},x_{3},x_{4}]_{\rho}:=-v(\det
R(\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3}),\varphi(x_{4}))).$

######  Theorem 7.3.

Let $\Lambda:=v(\mathbf{Q}(\rho))$ be the value group of the field
$\mathbf{Q}(\rho)$ generated over $\mathbf{Q}$ by the matrix coefficents of
$\rho$. Then

$[x_{1},x_{2},x_{3},x_{4}]_{\rho}\in\frac{1}{(8n)!}\Lambda$

for all $(x_{1},x_{2},x_{3},x_{4})\in H_{\Gamma}^{[4]}$.

###### Proof.

We might assume that $\mathbf{F}$ is the real closure of $\mathbf{Q}(\rho)$.
Let $(x_{1},x_{2},x_{3},x_{4})\in H_{\Gamma}^{[4]}$ and $\gamma_{i}$
hyperbolic with $(\gamma_{i})_{-}=x_{i}$. If $c_{\gamma_{1}}$,
$c_{\gamma_{2}}$, $c_{\gamma_{3}}$, $c_{\gamma_{4}}$ are the characteristic
polynomials of $\rho(\gamma_{1})$, $\rho(\gamma_{2})$, $\rho(\gamma_{3})$,
$\rho(\gamma_{4})$ then $c_{\gamma_{i}}\in\mathbf{Q}(\rho)[X]$. Since
$\mathbf{F}[i]$ is algebraically closed, $\rho(\gamma_{i})$ splits in
$\mathbf{F}[i]$. If $\mathbf{L}$ is the splitting field in $\mathbf{F}[i]$ of
$c_{\gamma_{1}}c_{\gamma_{2}}c_{\gamma_{3}}c_{\gamma_{4}}\in\mathbf{Q}(\rho)[X]$
then

$[\mathbf{L}:\mathbf{Q}(\rho)]\leq(8n)!\;.$

Observe that the field $\mathbf{L}$ depends on $\rho(\gamma_{1})$,
$\rho(\gamma_{2})$, $\rho(\gamma_{3})$, $\rho(\gamma_{4})$.

It is now easy to see that the Lagrangians
$\varphi(x_{i})\subset\mathbf{F}^{2n}$ are defined over
$\mathbf{L}\cap\mathbf{F}$, as a result we can represent them by
$\left\langle\begin{pmatrix}X_{i}\\\
\mathrm{Id}_{n}\end{pmatrix}\right\rangle$ with
$X_{i}\in\mathrm{Sym}_{n}(\mathbf{L}\cap\mathbf{F})$, which implies that $\det
R(\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3}),\varphi(x_{4}))\in(\mathbf{L}\cap\mathbf{F})^{\times}$.
We conclude using [Lan02, XII §4 Proposition 12] which says that the index of
$\Lambda$ in $v((\mathbf{L}\cap\mathbf{F})^{\times})$ is at most $(8n)!$. This
concludes the proof. ∎

In particular, if $\mathbf{Q}(\rho)$ has discrete valuation, we can assume, up
to rescaling the valuation, that the crossratio
$[x_{1},x_{2},x_{3},x_{4}]_{\rho}$ is integer valued. Theorem 1.4 is therefore
a direct application of Proposition 4.12.

##  8\. Examples of maximal framed representations

In this section we collect several interesting examples of maximal framed
representations over non-Archimedean real closed fields.

###  8.1. Ultralimits of representations and asymptotic cones

Let $(\rho_{k})_{k\geq 1}$ be a sequence of maximal representations into
$\operatorname{Sp}(2n,\mathbf{R})$ and $\omega$ a non-principal ultrafilter on
$\mathbf{N}$. This gives rise to a representation
$\rho_{\omega}\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{R}_{\omega})$, where
$\mathbf{R}_{\omega}$ is the field of hyperreals and
$\rho_{\omega}(\Gamma)\subset\operatorname{Sp}(2n,\mathcal{O}_{\sigma})$,
where the infinitesimal $\sigma$ is defined below. Denoting by
$\mathbf{R}_{\omega,\sigma}$ is the Robinson field, the representation
$\rho_{\omega,\sigma}$ obtained by composing $\rho_{\omega}$ with the
projection

$\operatorname{Sp}(2n,\mathbf{R}_{\omega})\to\operatorname{Sp}(2n,\mathbf{R}_{\omega,\sigma})\,,$

is a maximal framed representation of $\Gamma$ into
$\operatorname{Sp}(2n,\mathbf{R}_{\omega,\sigma})$, [BP17, Corollary 10.4];
its framing is defined on $\partial{\mathcal{H}}^{2}$, and Theorem 1.2
applies.

This construction is closely related to asymptotic cones, as we now recall.
Denoting by $d$ the $\operatorname{Sp}(2n,\mathbf{R})$-invariant Riemannian
distance on the Siegel $n$-space, we say that a sequence of scales
$(\lambda_{k})_{k\in\mathbf{N}}\in(\mathbf{R}_{>0})^{\mathbf{N}}$ is adapted
(to the sequence $(\rho_{k})_{k\in\mathbf{N}}$) if for one, and hence every,
finite generating set $S\subset\Gamma$

(14) $\lim_{\omega}\frac{\max_{\gamma\in
S}d\big{(}\rho_{k}(\gamma)\imath\mathrm{Id}_{n},\imath\mathrm{Id}_{n}\big{)}}{\lambda_{k}}<+\infty\,.$

We obtain then an action

${}^{\omega}\rho_{\lambda}\colon\Gamma\to\mathrm{Isom}({}^{\omega}\mathcal{X}_{\lambda})\,,$

on the asymptotic cone ${}^{\omega}\mathcal{S}_{\lambda}$ of the sequence of
pointed metric spaces given by
$(\mathcal{S}^{n}_{\mathbf{R}},\imath\mathrm{Id}_{n},\frac{d}{\lambda_{k}})$.

If we set $\sigma:=(e^{-\lambda_{k}})_{k\geq 1}\in\mathbf{R}_{\omega}$, then
the asymptotic cone ${}^{\omega}\mathcal{X}_{\lambda}$ can be identified with
the metric space $\mathcal{B}_{n}^{\mathbf{R}_{\omega,\sigma}}$ and, under
this identification, ${}^{\omega}\rho_{\lambda}$ corresponds to
$\rho_{\omega,\sigma}$ (see for example [Par12]).

###  8.2. Maximal representations in $\operatorname{SL}(2,\mathbf{F})$

Let $\mathbf{F}$ be a real closed field with an order compatible non-
Archimedean valuation, and let
$\mathcal{T}^{\mathbf{F}}\supset\mathcal{B}_{1}^{\mathbf{F}}$ be the
$\mathbf{R}$-tree associated to $\operatorname{SL}_{2}(\mathbf{F})$. Then
$\mathbb{P}^{1}(\mathbf{F})$ identifies with a subset of
$\partial_{\infty}\mathcal{T}^{\mathbf{F}}$ and the restriction to
$\mathbb{P}^{1}(\mathbf{F})$ of the crossratio of
$\partial_{\infty}\mathcal{T}^{\mathbf{F}}$ is the standard crossratio
$[\,\cdot\,,\,\cdot\,,\cdot\,,\,\cdot\,]_{\mathbf{F}}$ in
$\mathbb{P}^{1}(\mathbf{F})$.

Hence any representation $\rho\colon\Gamma\to\operatorname{SL}(2,\mathbf{F})$
with framing $\varphi\colon{H_{\Gamma}}\to\mathbb{P}^{1}(\mathbf{F})$, gives a
framed action on $\mathcal{T}^{\mathbf{F}}$. Note that the associated
crossratio
$[x_{1},x_{2},x_{3},x_{4}]_{\varphi}=[\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3}),\varphi(x_{4})]_{\mathbf{F}}$
is positive if the framing $\varphi$ is maximal.

Proposition 5.8 implies the following.

######  Theorem 8.1.

Let $\mathbf{F}$ be a real closed field with an order compatible non-
Archimedean valuation. Let
$\rho\colon\Gamma\to\operatorname{SL}(2,\mathbf{F})$ be a representation with
a maximal framing $\varphi\colon{H_{\Gamma}}\to\mathbb{P}^{1}(\mathbf{F})$.
Denote by $\mu_{\rho}$ the geodesic current associated to the positive
crossratio induced by $\varphi$ on $X$. Then $\mu_{\rho}$ corresponds to a
measured lamination, and there is a $\Gamma$-equivariant isometric embedding

$\mathcal{V}(\mu_{\rho})\hookrightarrow\mathcal{T}^{\mathbf{F}}.$

In particular, for all hyperbolic $\gamma\in\Gamma$,
$\ell(\rho(\gamma))=i(\mu_{\rho},\gamma).$

###  8.3. Unipotent representations of the thrice punctured sphere

Let $\Gamma<\operatorname{PSL}(2,\mathbf{R})$ be the (unique up to
conjugation) lattice such that $\Gamma\backslash{\mathcal{H}}^{2}$ is the
thrice punctured sphere. Then $\Gamma$ admits a presentation

$\Gamma=\left\langle c_{1},c_{2},c_{3}:\,c_{3}c_{2}c_{1}\right\rangle\,,$

where $c_{1},c_{2},c_{3}$ are parabolic elements representing the three
inequivalent cusps of $\Gamma$. Already in this elementary example we are able
to illustrate interesting features. For every $\alpha\in\mathbf{R}$ we
construct maximal framed representations
$\rho_{\alpha}\colon\Gamma\to\operatorname{Sp}(4,\mathcal{H}(\mathbf{R}))$,
where $\mathcal{H}(\mathbf{R})$ is the Hahn field with exponents $\mathbf{R}$
(see Example 6.2(4)), that have the following properties:

  1. (1)

for $\alpha\leq 1/2$ the corresponding length functions $\gamma\mapsto
L(\rho_{\alpha}(\gamma))$ are not proportional and hence the corresponding
currents $\mu_{\rho_{\alpha}}$ are distinct in the space of projectivized
currents;

  2. (2)

for $\alpha\in\mathbf{Q}$ the associated geodesic current
$\mu_{\rho_{\alpha}}$ is a multicurve.

To this end we use the explicit coordinates obtained by Strubel on the set of
$\operatorname{Sp}(2n,\mathbf{R})$-conjugacy classes of maximal
representations of $\Gamma$ into $\operatorname{Sp}(2n,\mathbf{R})$. Namely,
let

$\overline{B}:=\\{A\in\operatorname{GL}(n,\mathbf{R}):\,\mathrm{spec}(A)\subset\\{z\in\mathbf{C}:\,|z|\leq
1\\}\\}$

and

$R=\\{(X_{1},X_{2},X_{3})\in\overline{B}^{3}:\,X_{3}\,{}^{t}X_{2}^{-1}X_{1}\text{
is symmetric positive definite}\\}\,.$

Then, given $X:=(X_{1},X_{2},X_{3})\in R$, the formulas

$\displaystyle\rho_{X}(c_{1})$ $\displaystyle=\begin{pmatrix}X_{1}&0\\\
X_{1}+X_{2}^{-1}{}^{t}X_{3}&{}^{t}X_{1}^{-1}\end{pmatrix}$
$\displaystyle\rho_{X}(c_{3})$
$\displaystyle=\begin{pmatrix}{}^{t}X_{3}^{-1}&-{}^{t}X_{3}^{-1}-X_{1}^{-1}{}^{t}X_{2}\\\
0&X_{3}\end{pmatrix}$

determine a representation $\Gamma\to\operatorname{Sp}(2n,\mathbf{R})$ that is
maximal [Str15, Theorem 2]. Moreover every maximal representation of $\Gamma$
into $\operatorname{Sp}(2n,\mathbf{R})$ is conjugate to a $\rho_{X}$ for $X\in
R$ and $\operatorname{Sp}(2n,\mathbf{R})$-conjugacy classes of maximal
representations correspond to $\mathrm{O}(n)$-conjugacy classes in $R$ for the
diagonal conjugation action of $\mathrm{O}(n)$.

If now $\mathbf{F}$ is a real closed field and

$\overline{B}_{\mathbf{F}}=\big{\\{}A\in\operatorname{GL}(n,\mathbf{F}):\,\mathrm{spec}(A)\subset\\{z\in\mathbf{F}(\sqrt{-1}):\,|z|\leq
1\\}\big{\\}}$

and

$R_{\mathbf{F}}:=\\{(X_{1},X_{2},X_{3})\in\overline{B}_{\mathbf{F}}^{3}:\,X_{3}\,{}^{t}X_{2}^{-1}X_{1}\text{
is symmetric positive definite}\\}\,,$

the following gives a source of maximal framed representations over any real
closed field $\mathbf{F}$:

######  Proposition 8.2.

For every $X:=(X_{1},X_{2},X_{3})\in R_{\mathbf{F}}$ the formulas for
$\rho_{X}(c_{1})$ and $\rho_{X}(c_{3})$ define a maximal framed representation
$\rho_{X}\colon\Gamma\to\operatorname{Sp}(2n,\mathbf{F})$.

The proof is beyond the scope of this paper. Let us just mention that it is an
easy application of the Tarski–Seidenberg principle (see [BCR98, Proposition
5.1.3]) and the fact that $R$ parametrizes a semi-algebraic subset of
$\mathrm{Hom}(\Gamma,\operatorname{Sp}(2n,\mathbf{R}))$.

We are interested in the case in which $\rho_{X}(c_{1})$, $\rho_{X}(c_{2})$
and $\rho_{X}(c_{3})$ are all unipotent, which is equivalent to
$X_{1},-X_{2},X_{3}$ being unipotent. This is never the case if $n=1$ as one
can see from the above formulas, while already for
$\operatorname{Sp}(4,\mathbf{R})$ there are interesting examples with
unipotent boundary holonomy. We restrict here to the subset of $R$ consisting
of triple $(X_{1},X_{2},X_{3})$ such that $X_{1},-X_{2},X_{3}$ are unipotent
and $X_{3}{}^{t}X_{2}^{-1}X_{1}=\mathrm{Id}$. The quotient by
$\mathrm{O}(2)$-conjugation of such triples can be parametrized by

$\left\\{\begin{pmatrix}1&\frac{4}{x}\\\
0&1\end{pmatrix},\begin{pmatrix}-3+y&-x\\\
\frac{(y-2)^{2}}{x}&1-y\end{pmatrix},\begin{pmatrix}1+y&\frac{y^{2}}{x}\\\
-x&1-y\end{pmatrix},x>0,y\in\mathbf{R}\right\\}\,.$

The corresponding matrices are then

$\displaystyle\rho_{X}(c_{1})=$ $\displaystyle\begin{pmatrix}X_{1}&0\\\
X_{1}+X_{2}^{-1}{}^{t}X_{3}&{}^{t}X_{1}^{-1}\end{pmatrix}=\begin{pmatrix}1&\frac{4}{x}&0&0\\\
0&1&0&0\\\ 2&\frac{4}{x}&1&0\\\ -\frac{4}{x}&2&-\frac{4}{x}&1\end{pmatrix}$
$\displaystyle\rho_{X}(c_{3})=$
$\displaystyle\begin{pmatrix}{}^{t}X_{3}^{-1}&-{}^{t}X_{3}^{-1}-X_{1}^{-1}{}^{t}X_{2}\\\
0&X_{3}\end{pmatrix}=\begin{pmatrix}1-y&x&-2&-x-\frac{y^{2}}{x}\\\
-\frac{y^{2}}{x}&1+y&x+\frac{y^{2}}{x}&-2\\\ 0&0&1+y&\frac{y^{2}}{x}\\\
0&0&-x&1-y\end{pmatrix}\,.$

The above formulas allow us to consider $\rho_{X}$ as a representation of
$\Gamma$ with coefficients in the ring $\mathbf{R}[x,\frac{1}{x},y]$. Now for
every $\alpha\in\mathbf{R}$ we consider the ring morphism of
$\mathbf{R}[x,\frac{1}{x},y]$ into the Hahn field $\mathcal{H}(\mathbf{R})$
defined by sending $x$ to $x$ and $y$ to $x^{\alpha}$. In this way we obtain
for every $\alpha\in\mathbf{R}$ a representation
$\rho_{\alpha}\colon\Gamma\to\operatorname{Sp}(4,\mathcal{H}(\mathbf{R}))$. It
is easy to verify that the triple $X=(X_{1},X_{2},X_{3})$ with $y=x^{\alpha}$
is in $R_{\mathcal{H}(\mathbf{R})}$ and it follows from Proposition 8.2 that
$\rho_{\alpha}$ is maximal framed. For the computation of the length function
$L$ it is not difficult to see that if $g\in\operatorname{Sp}(4,\mathbf{F})$,
where $\mathbf{F}$ is real closed non-Archimedean with an order compatible
valuation, then

$L(g)=-v(T(g))\,.$

where $T(g)=(\mathrm{tr}g)^{2}-\mathrm{tr}g^{2}-4$. In our case we obtain

$T(\rho_{\alpha}(c_{1}^{-1}c_{3}))=4(4x^{2}+32x^{-4+4\alpha}+(18+8x^{2\alpha})+4x^{-2}(16+12x^{2\alpha}+x^{4\alpha}))\,$

and

$T(\rho_{\alpha}(c_{1}^{-1}c_{2}))=4(50+4x^{2}+8x^{2\alpha}+4x^{-2}(16+12x^{2\alpha}+x^{4\alpha}))$

so

$v(T(\rho_{\alpha}(c_{1}^{-1}c_{3})))=\min(-2,-4+4\alpha,-2+2\alpha)=\begin{cases}-2&\alpha\geq\frac{1}{2}\\\
-4+4\alpha&\alpha\leq\frac{1}{2}\end{cases}$

and

$v(T(\rho_{\alpha}(c_{1}^{-1}c_{2})))=\min(-2,-2+4\alpha,-2+2\alpha)=\begin{cases}-2&\alpha\geq
0\\\ -2+4\alpha&\alpha\leq 0\,.\end{cases}\,$

We deduce that for $\alpha\leq 1/2$ the length functions $\gamma\mapsto
L(\rho_{\alpha}(\gamma))$ are distinct even when considered up to positive
scaling. It is easy to verify that

$\mathbf{Q}(\rho_{\alpha})=\begin{cases}\mathbf{Q}(x^{\alpha})&\text{ if
}\alpha\in\mathbf{Q}\smallsetminus\\{0\\}\\\ \mathbf{Q}(x)&\text{ if
}\alpha=0\\\ \mathbf{Q}(x,x^{\alpha})&\text{ if }\alpha\notin\mathbf{Q}\,.\\\
\end{cases}$

As a result, the image of the valuation is

$v(\mathbf{Q}(\rho_{\alpha}))=\begin{cases}\alpha\mathbf{Z}&\text{ if