Weyl chamber length compactification of the $\textrm{PSL}(2,{\mathbb{R}})\times \textrm{PSL}(2,{\mathbb{R}})$ maximal character variety

2.1. Weyl chamber-valued and ${\mathbb{R}}^n$ -valued metrics

Given a finite reflection group $({\mathfrak{a}},W)$ and a fixed model closed Weyl chamber ${\overline{\mathfrak a}^+}$ , we define a ${\overline{\mathfrak a}^+}$ -valued pseudometric on a space $X$ as a function $\overrightarrow{d} \colon X \times X \to{{\overline{\mathfrak a}^+}}$ satisfying $\overrightarrow{d}(x,x)=0$ and

1. Triangular inequality: $\overrightarrow{d}(x,z)\leq \overrightarrow{d}(x,y)+\overrightarrow{d}(y,z)$ ;

2. $\text{opp}$ -symmetry: $\overrightarrow{d}(y,x)=\overrightarrow{d}(x,y)^{\text{opp}}$ ,

where $\text{opp}$ is the opposition involution on $\mathfrak{a}$ , and $\mathfrak{a}$ is endowed with the partial order with positive cone the Euclidean dual $({{\overline{\mathfrak a}^+}})^*$ of ${\overline{\mathfrak a}^+}$ . We will call $\overrightarrow{d}$ a ${\overline{\mathfrak a}^+}$ -valued metric if in addition it is separated, namely $\overrightarrow{d}(x,y)=0$ implies $x=y$ .

In this article, we are interested only in the reflection group associated to the semisimple Lie group $G=\textrm{PSL}(2,{\mathbb{R}})^n$ , which is ${\mathfrak{a}}={\mathbb{R}}^n$ with Weyl group $W=\{\pm{\textrm{id}}_{\mathbb{R}}\}^n$ . The model closed Weyl chamber is ${{\overline{\mathfrak a}^+}}={\mathbb{R}}_+^n$ . In this case, the involution opposition is trivial, $({{\overline{\mathfrak a}^+}})^*={{\overline{\mathfrak a}^+}}={\mathbb{R}}_+^n$ so a ${\overline{\mathfrak a}^+}$ -valued (pseudo)metric amounts simply to a $n$ -tuple

\begin{align*}\overrightarrow {d}=(d_i)_{i=1,\ldots,n}\end{align*}

of usual $\mathbb{R}$ -valued (pseudo)metric $d_i$ . This gives rise to the equivalent notion of a ${\mathbb{R}}^n$ -(pseudo)metric:

Observe that even if $(X,\overrightarrow{d})$ is an ${\mathbb{R}}^n$ -metric space, it is possible that none of the pseudo-metrics $d_i$ is separated.

Example 2.2. Examples of ${\mathbb{R}}^n$ -metric spaces are products

\begin{align*}X=X_1\times \ldots \times X_n\end{align*}

of ordinary metric spaces, as well as subspaces thereof with the induced metric. Another example that will play an important role in our paper are the ${\mathbb{R}}^2$ -metric naturally induced on the universal cover of a half translation surface, see Section 5.1 for details. Another relevant example is given by simplicial trees with ${\mathbb{R}}^2$ -valued edge lengths. This is discussed, and generalized, in Section 3.5.

In our case of interest,Footnote 2 it is easy to prove that, for any $W$ -invariant norm $N\colon{\mathfrak{a}}={\mathbb{R}}^n\to{\mathbb{R}}_+$ , $N$ is not decreasing on each variable on ${{\overline{\mathfrak a}^+}}={\mathbb{R}}_+^n$ , that is, for the partial order introduced above. As a result, a ${\overline{\mathfrak a}^+}$ -valued (pseudo)metric $\overrightarrow{d}$ on $X$ induces an $\mathbb{R}$ -valued (pseudo)metric by setting

\begin{align*}d_N(x,y)\;:\!=\;N(\overrightarrow {d}(x,y)) \;.\end{align*}

Note that $\overrightarrow{d}(x,y)=0$ if and only if $d_N(x,y)=0$ , in particular $\overrightarrow{d}$ is separated if and only if $d_N$ is separated.

Norms $N\colon{\mathbb{R}}^n\to{\mathbb{R}}$ of particular interest to us are the $\ell ^2$ and $\ell ^1$ norms, which we denote, respectively, by $\|\cdot \|_2$ and $\|\cdot \|_1$ . They give the associated Euclidean (pseudo)metric

\begin{align*}d_{\|\cdot \|_2}(x,y)\;:\!=\;\sqrt {\sum _{i=1}^n d_i(x,y)^2}\end{align*}

and $\ell ^1$ -(pseudo)metric

\begin{align*}d^1(x,y)\;:\!=\;\sum _{i=1}^n d_i(x,y)\end{align*}

on $X$ . We will also denote $d_{\|\cdot \|_2}$ by $\lVert \overrightarrow{d} \rVert _2$ and $d^1$ by $\| \overrightarrow{d} \|_1$ .

2.2. The Weyl chamber valued length function on $\textrm{PSL}(2,{\mathbb{R}})^n$

The symmetric space associated to $\textrm{PSL}(2,{\mathbb{R}})^n$ is the product $X=(\mathcal H^2)^n$ of $n$ copies of the hyperbolic plane $\mathcal H^2$ . On $X$ the natural Weyl-chamber valued distance $\overrightarrow{d} \colon X \times X \to{\mathbb{R}}_+^n$ is simply the product distance

\begin{align*}\overrightarrow {d}(x,y)=(d(x_i,y_i)_i)_{i=1,\ldots,n}\;.\end{align*}

The Weyl chamber length (or ${\mathbb{R}}^n$ -length) of $g=(g_i)_{i=1,\ldots,n}\in \textrm{PSL}(2,{\mathbb{R}})^n$ can be defined as

(1.1) \begin{align} \overrightarrow{L}(g)\;:\!=\;\inf _{x\in X}\overrightarrow{d}(x,gx) \end{align}

where the infimum is considered with respect to the partial order on ${\mathbb{R}}^n$ with positive cone ${\mathbb{R}}_+^n$ . As the metric space $X$ is a product, it boils down to taking the list of the usual lengths in each factor:

\begin{align*}\overrightarrow {L}(g)=({L}(g_i))_{i=1,\ldots,n}.\end{align*}

Here ${L}(g)\;:\!=\;\inf _{x\in \mathcal H^2}d(x,gx)$ is the usual translation length of $g$ on the hyperbolic plane $\mathcal H^2$ . This ensures in particular that the infimum in (1.1) exists.

The length function of a representation $\rho \colon \Gamma \to \textrm{PSL}(2,{\mathbb{R}})$ is

\begin{align*}L_\rho \;:\!=\;{L}\circ \rho \colon \Gamma \to {\mathbb {R}}_+ \;.\end{align*}

Similarly, the Weyl chamber length function (or ${\mathbb{R}}^n$ -length function) of a representation $\rho \colon \Gamma \to \textrm{PSL}(2,{\mathbb{R}})^n$ is

\begin{align*}\overrightarrow {L}_\rho \;:\!=\;\overrightarrow {L}\circ \rho \colon \Gamma \to {\mathbb {R}}_+^n \;. \end{align*}

Of course when $\rho =(\rho _i)_{i=1,\ldots,n}$ , it holds $\overrightarrow{L}_\rho =(L_{\rho _i})_{i=1,\ldots,n}$ .

2.3. The Weyl chamber length compactification

Here we define the Weyl chamber length compactification constructed in [Reference Parreau25], which has a simpler construction in our context. We endow the space $({\mathbb{R}}_+^n)^\Gamma$ of functions $\overrightarrow{L}\colon \Gamma \to{\mathbb{R}}_+^n$ with the topology of pointwise convergence, and the space ${\mathbb{P}}(({\mathbb{R}}_+^n)^\Gamma )$ of homothety classes of nonzero functions $\overrightarrow{L}$ with the quotient topology. The map

\begin{align*}\begin {array}{c@{\quad}c@{\quad}c@{\quad}c} \overrightarrow {\mathcal {L}}:&{\textrm {Hom}}(\Gamma,\textrm {PSL}(2,{\mathbb {R}})^n) &\to & ({\mathbb {R}}_+^n)^\Gamma \\[5pt] &\rho &\mapsto & \overrightarrow {L}\circ \rho \end {array}\end{align*}

induces a continuous map

\begin{align*}{\mathbb {P}}\overrightarrow {\mathcal {L}}\colon \mathcal {T}^n \to {\mathbb {P}}(({\mathbb {R}}_+^n)^\Gamma )\end{align*}

equivariant with respect to the mapping class group action. The injectivity of $\mathcal{T} \to{\mathbb{P}}{\mathbb{R}}_+^\Gamma$ implies that the map ${\mathbb{P}}\overrightarrow{\mathcal{L}}$ is injective as well. The Weyl chamber length compactification $\overline{\mathcal{T}^n}^{\textrm{WL}}$ of $\mathcal{T}^n$ is by definition the closure of its image (which is a compact set). We denote

\begin{align*}\partial ^{\textrm {WL}}(\mathcal {T}^n)\;:\!=\;\overline {\mathcal {T}^n}^{\textrm {WL}}\!\!-{\mathbb {P}}\overrightarrow {\mathcal {L}}(\mathcal {T}^n)\subset {\mathbb {P}}(({\mathbb {R}}_+^n)^\Gamma )\end{align*}

its boundary.

2.4. Other length compactifications

Recall from Section 2.1 that any $W$ -invariant norm $N\colon{\mathbb{R}}^n\to{\mathbb{R}}$ induces a metric $d_N\;:\!=\;N(\overrightarrow{d})$ on $X=(\mathcal H^2)^n$ , and thus a corresponding length function, the $N$ -length function $L_{d_N}$ . We use this to construct the $N$ -length compactification $\overline{\mathcal{T}^n}^{N}$ of $\mathcal{T}^n$ , associated to $L_{d_N}$ .

It is easy to verify that, since $N$ is not decreasing on each variable on ${{\overline{\mathfrak a}^+}}={\mathbb{R}}_+^n$ , and $\mathcal{T}^n$ is a product, it holds $L_{d_N}(g)=N(\overrightarrow{L}(g))$ . The Weyl chamber length compactification then dominates naturally $\overline{\mathcal{T}^n}^{N}$ : the restriction of the natural map

\begin{align*} \begin {array}[t]{rll} ({\mathbb {R}}_+^n)^\Gamma &\to & {\mathbb {R}}_+^\Gamma \\[5pt] \overrightarrow {L}&\mapsto & N\circ \overrightarrow {L} \end {array} \end{align*}

induces a continuous $\textrm{Out}(\Gamma )$ -equivariant surjective map

\begin{align*}\overline {\mathcal {T}^n}^{\textrm {WL}} \to \overline {\mathcal {T}^n}^{N} \end{align*}

restricting to identity on $\mathcal{T}^n$ .

3.1. Measured foliations

Let $\mathcal{F}$ be a transverse measured foliation on a topological surface $\Sigma$ with fundamental group $\Gamma$ . We denote by $i(\mathcal {F},c)$ the measure of a path $c\subset \Sigma$ with respect to $\mathcal{F}$ . The intersection of $\mathcal{F}$ with $\gamma \in \Gamma$ is defined as

\begin{align*}i_{\mathcal {F}}(\gamma )\;:\!=\;\inf _c i(\mathcal {F},c)\end{align*}

where the infimum is taken over all closed loops $c$ transverse to $\Lambda$ and freely homotopic to $\gamma$ . We denote by

\begin{align*}I_{\mathcal {F}}\;:\!=\;i(\mathcal {F},\cdot )\colon \Gamma \to {\mathbb {R}}_+\end{align*}

the corresponding intersection function.

This may be reformulated in terms of pseudometrics. We denote by $d_{\mathcal{F}}$ the pseudometric on the universal cover $\widetilde{\Sigma }$ of $\Sigma$ associated with the lift $\widetilde{\mathcal{F}}$ of $\mathcal{F}$ : it is defined by the formula

\begin{align*}d_{\mathcal {F}}(x,y) \;:\!=\;\inf _c i(\widetilde {\mathcal {F}}, c)\end{align*}

where the infimum is taken on paths $c$ joining $x$ to $y$ in $\widetilde{\Sigma }$ and transverse to $\widetilde{\mathcal{F}}$ . The length function $L_d\in ({\mathbb{R}}_+)^\Gamma$ of a pseudometric $d$ on a $\Gamma$ -space $X$ is defined as

\begin{align*}L_d(\gamma )\;:\!=\;\inf _{x\in X} d(x,\gamma x) \; .\end{align*}

By definition, the length function of $d_{\mathcal{F}}$ on $\widetilde{\Sigma }$ is the intersection function of $\mathcal{F}$ :

\begin{align*}L_{d_{\mathcal {F}}}=I_{\mathcal {F}}.\end{align*}

The dual tree of a measured foliation $\mathcal{F}$ is defined as the quotient metric space $T(\mathcal{F})\;:\!=\;\widetilde{\Sigma }/d_{\mathcal{F}}$ of $\widetilde{\Sigma }$ by the pseudometric $d_{\mathcal{F}}$ . The tree $T(\mathcal{F})$ inherits an action of $\Gamma$ by isometries, with associated length function

\begin{align*}L_{T(\mathcal{F})}=I_{\mathcal {F}} \;.\end{align*}

3.2. Measured geodesic laminations

Let $\lambda =(\Lambda,\nu )$ be a measured lamination on $\Sigma$ . As for measured foliations, we denote by $i(\lambda, c)$ the $\lambda$ -measure of a path $c\colon [0,1] \to \Sigma$ transverse to $\Lambda$ . The intersection of $\lambda$ with $\gamma \in \Gamma$ is defined as

\begin{align*}i(\lambda, \gamma )\;:\!=\;\inf _c i(\lambda,c)\end{align*}

where the infimum is taken over all closed loops $c$ transverse to $\Lambda$ and freely homotopic to $\gamma$ , and we denote by

\begin{align*}I_\lambda \;:\!=\;i(\lambda,\cdot )\colon \Gamma \to {\mathbb {R}}_+\end{align*}

the corresponding intersection function.

We denote by $d_\lambda$ the pseudometric on $\widetilde{\Sigma }$ associated with $\lambda$ : this is the pseudometric defined on $\widetilde{\Sigma }-\widetilde{\Lambda _0}$ , where $\widetilde{\Lambda _0}$ is the set of atomicFootnote 3 leafs of the lift $\tilde{\lambda }=(\widetilde{\Lambda },\nu )$ of $\lambda$ , by the formula

\begin{align*}d_\lambda (x,y) \;:\!=\;\inf _c i(\tilde {\lambda }, c)\end{align*}

where the infimum is taken on paths $c$ joining $x$ to $y$ in $\widetilde{\Sigma }$ and transverse to $\widetilde{\Lambda }$ . By definition, the length function of $d_\lambda$ on $\widetilde{\Sigma }-\widetilde{\Lambda _0}$ is the intersection function of $\lambda$ :

\begin{align*}L_{d_\lambda }=I_\lambda .\end{align*}

We refer to [Reference Kapovich14] for the classical correspondence between measured foliations and measured geodesic laminations on hyperbolic surfaces.

3.3. Geodesic currents

We refer to [Reference Bonahon2] or [Reference Martelli15, Section 8] for the background material in the case of closed geodesic surfaces, and for instance to [Reference Burger, Iozzi, Parreau and Pozzetti4] for the generalization to finite type surfaces.

A geodesic current on a finite type hyperbolic surface $\Sigma =\mathcal H^2/\Gamma$ is a flip-invariant $\Gamma$ -invariant positive Radon measure on the space of unoriented, unparametrized geodesics of $\mathcal H^2$ , that may be identified with the space

\begin{align*} \mathcal {G}(\mathcal H^2)\;:\!=\;\{(x,y)\in (\partial \mathcal H^2)^2:\,x\neq y\} \end{align*}

of distinct pairs of points in the boundary at infinity $\partial \mathcal H^2$ of $\mathcal H^2$ . A basic example is the current $\delta _c$ associated to a closed geodesic $c$ in $\Sigma$ , which is defined as the sum of the Dirac masses on the lifts of $c$ to $\mathcal H^2$ . Recall that the Bonahon intersection $i(\mu,\nu )$ of two geodesic currents $\mu,\nu$ is defined as the $(\mu \times \nu )$ -measure of any Borel fundamental domain for the $\Gamma$ -action on the space $\mathcal{DG}(\mathcal H^2)\subset \mathcal{G}(\mathcal H^2)\times \mathcal{G}(\mathcal H^2)$ of pairs of transverse geodesics $(g,h)$ .

We refer for instance to [Reference Martelli15, Section 8.3.4] for the bijective correspondence $\lambda \mapsto \mu _ \lambda$ between measured geodesic laminations $\lambda =(\Lambda,\nu )$ of full support $\Lambda$ and geodesic currents $\mu$ on $\Sigma$ with $i(\mu,\mu )=0$ , equivalently currents $\mu$ such that no two geodesics in the support of $\mu$ intersect transversally. The geodesic current $\mu _\lambda$ has support $\Lambda$ , and, for each geodesic arc $c$ transverse to $\Lambda$ , the restriction of $\mu _\lambda$ to the set of geodesics $g$ of $\Lambda$ intersecting $c$ is the pullback of the $\lambda$ -measure on $c$ by the map $g \mapsto g\cap c$ . The notions of intersection then coincide as

\begin{align*} i(\mu _\lambda,\delta _c)=i(\lambda,c)\,, \end{align*}

for all closed geodesic $c$ in $\Sigma$ . Note that the union of two measured geodesic laminations with disjoint support corresponds to the sum of the associated currents. From now on, we will freely identify $\lambda$ with $\mu _\lambda$ whenever convenient.

Given a geodesic current $\mu$ on $\Sigma$ and a geodesic subsurface $\Sigma^{\prime}$ of $\Sigma$ , we denote by $\mu_{|{\Sigma^{\prime}}}$ the restriction of $\mu$ to the subsurface $\Sigma^{\prime}$ , namely the geodesic current

\begin{align*}\mu _{{\Sigma^{\prime}}}\;:\!=\;\chi _{\mathcal {G}(\Sigma^{\prime})}\mu,\end{align*}

where $\chi _{\mathcal{G}(\Sigma^{\prime})}$ is the characteristic function of the set $\mathcal{G}(\Sigma^{\prime})$ of geodesics whose projection lies in $\Sigma^{\prime}$ . In general, a geodesic current of full support might restrict to the zero current on a proper subsurface (this is the case when $\mu$ is the Liouville current of a hyperbolic structure), but if, for every boundary component $c$ of $\Sigma^{\prime}$ , $i(\mu,\delta _c)=0$ , then for every $\gamma \in \pi _1(\Sigma^{\prime})$ , $i(\mu,\delta _\gamma )=i(\mu_{|\Sigma^{\prime}},\delta _\gamma )$ [Reference Burger, Iozzi, Parreau and Pozzetti5, Proposition 4.13].

3.4. Dual tree of a measured geodesic lamination

Let $\lambda =(\Lambda,\nu )$ be a measured geodesic lamination on an hyperbolic surface $\Sigma$ . We now recall the construction of the associated dual $\mathbb{R}$ -tree. We follow the construction of [Reference Kapovich14, Section 11.12].

We first get rid of the atoms blowing-up along atomic leafs: for each isolated leaf $c$ of $\Lambda$ , cut $\Sigma$ along $c$ and insert an annulus $B(c)=c\times [0,1]$ , foliated by the parallel circles $c\times{t}$ . We endow $\Sigma ^b$ with the locally CAT(0) metric $m^b$ equal to the original metric $m$ of $\Sigma$ outside the annuli $B(c)$ and to the flat metric on $B(c)$ . This gives a locally CAT(0) surface $(\Sigma ^b,m^b)$ homeomorphic to $\Sigma$ , with a geodesic lamination $\Lambda ^b$ . We call $(\Sigma ^b,\Lambda ^b)$ the blow-up of $(\Sigma,\Lambda )$ . The blow-up of the transverse measure $\nu$ on $\Lambda$ is the non-atomic transverse measure $\nu ^b$ on $\Lambda ^b$ , obtained from $\nu$ by giving to each foliated annulus $B(c)$ the transverse measure $\nu (c)dt$ on $[0,1]$ . The pseudometric $d_{\lambda ^b}$ associated to the measured lamination $\lambda ^b\;:\!=\;(\Lambda ^b,\nu ^b)$ is then a continuous and everywhere-defined path pseudometric on $\widetilde{\Sigma ^b}$ . The $\mathbb{R}$ -tree dual to $\lambda$ is defined as the quotient metric space $T(\lambda )\;:\!=\;\widetilde{\Sigma ^b}/d_\lambda ^b$ , whose metric will be denoted by $d_\lambda$ .

It is easy but crucial to see that the geodesics of $T(\lambda )$ are the projections of the $m^b$ -geodesics of $\widetilde{\Sigma ^b}$ . In the following lemma, and in the rest of the article, when we write geodesic we mean a path $t\mapsto c(t)$ that is additive for the distance $d$ , namely such that $d(c(t_1),c(t_3))=d(c(t_1),c(t_2))+d(c(t_2),c(t_3))$ for all $t_1\leq t_2\leq t_3$ . Observe that $c$ may then be constant on some subset of its domain of definition and needs not be parametrized at constant speed.

Proof. Let $x,y$ be points of $\widetilde{\Sigma ^b}$ and $c\colon [0,1]\to \widetilde{\Sigma ^b}$ be the constant speed geodesic segment from $x$ to $y$ . As the leafs of $\widetilde{\Lambda }^b$ are geodesics, and CAT(0) spaces are uniquely geodesic, the path $c$ cannot cross twice the same leaf of $\widetilde{\Lambda }^b$ , hence $i(\lambda ^b,c)$ is minimal and $d_{\lambda ^b}(x,y)=i(\lambda ^b,c)$ . Denote by $p\colon \widetilde{\Sigma ^b}\to T(\lambda )$ the canonical projection. The projection $\overline{c}=p\circ c$ of $c$ in the dual tree $T(\lambda )$ does not backtrack, hence is a geodesic.

The tree $T(\lambda )$ inherits an action of $\Gamma$ by isometries, with associated length function

\begin{align*}L_{T(\lambda )}=I_\lambda \;.\end{align*}

Note that when $\Sigma$ is closed the tree $T(\lambda )$ is then minimal for the action of $\Gamma$ , namely there is no invariant proper subtree. Indeed an invariant proper subtree will lift as a proper closed invariant convex subset in $\widetilde{\Sigma ^b}$ , and taking the closure in the CAT(0) compactification we will then obtain a proper closed invariant subset of the boundary at infinity $\partial _\infty \widetilde{\Sigma ^b} \simeq \partial \mathcal H^2$ , which is impossible.

Note that the tree $T(\lambda )$ is essentially determined by its length function ; more generally, we will use the length rigidity of actions on minimal trees with length functions in $\mathcal{ML}$ : It is easily seen that for $\lambda \neq 0$ the action of $\Gamma$ on $T(\lambda )$ has no global fixed point in $\partial _\infty T$ , where $\partial _\infty T$ denotes the boundary at infinity of $T$ . This in fact depends only on the length function, since having a fixed point at infinity is equivalent to the length function being of the form $\gamma \mapsto |h(\gamma )|$ , where $h:\Gamma \to{\mathbb{R}}$ is a morphism [Reference Culler and Morgan7, Corollary 2.3]. In particular, all minimal $\Gamma$ -trees with nonzero length function in $\mathcal{ML}$ have no fixed point at infinity, in particular are semisimple and not shifts in the sense of [Reference Culler and Morgan7]. It follows then from the length rigidity for minimal semisimple $\Gamma$ -trees [Reference Culler and Morgan7, Theorem 3.7] that if $T$ and $T'$ are any two minimal $\Gamma$ -trees with the same length function in $\mathcal{ML}$ , then there is a unique equivariant isometry $T\to T'$ .

3.5. The ${\mathbb{R}}^n$ -tree dual to a $n$ -measured geodesic lamination

Let $n\in{\mathbb{N}}$ .

Observe that, while we only assume that $d_i$ are pseudodistances, we require that their sum $d^1$ is a distance, namely it separates points.

When $\Sigma$ is endowed with a locally CAT(0) metric, a $n$ -measured geodesic lamination $\overrightarrow{\lambda }=(\Lambda,\overrightarrow{\nu })$ is called geodesic when $\Lambda$ is geodesic.

When $\Sigma$ is a closed hyperbolic surface, seeing measured geodesic laminations as geodesic currents, a $n$ -tuple $(\lambda _1,\ldots, \lambda _n)$ of measured geodesic laminations is parallel if and only if

\begin{align*}i(\lambda _i, \lambda _j)=0 \text { for all } i,j\end{align*}

or equivalently that

\begin{align*}\lambda \;:\!=\;\sum _{i=1}^n\lambda _i\text { is a measured lamination.}\end{align*}

Let $\overrightarrow{\lambda }=(\Lambda,\overrightarrow{\nu })$ be a $n$ -measured lamination on a closed hyperbolic surface $\Sigma$ . We now construct the associated dual ${\mathbb{R}}^n$ -tree, adapting the construction of Section 3.4.

We take the $\mathbb{R}$ -tree $T(\lambda )\;:\!=\;\widetilde{\Sigma ^b}/d_\lambda ^b$ dual to the measured lamination $\lambda =(\Lambda,\nu )$ where $\nu =\sum _i\nu _i$ .

Recall from Section 3.4 that $(\Sigma ^b,\Lambda ^b)$ is the blow-up of $(\Sigma,\Lambda )$ , which is a locally CAT(0) surface $(\Sigma ^b,m^b)$ homeomorphic to $\Sigma$ , with a geodesic lamination $\Lambda ^b$ , and that $\lambda ^b=(\Lambda ^b,\nu ^b)$ where $\nu ^b$ is the non-atomic transverse measure on $\Lambda ^b$ obtained by blowing up $\nu$ , and that $d_{\lambda ^b}$ the associated continuous path pseudometric on $\widetilde{\Sigma ^b}$ .

Taking the blow-ups $\nu ^b_i$ of the transverse measure $\nu _i$ , we obtained a $n$ -tuple $\overrightarrow{\nu }^b\;:\!=\;(\nu ^b_i)_i$ of non-atomic transverse measures on $\lambda ^b$ (not necessarily of full support), namely a $n$ -measured lamination $\overrightarrow{\lambda }^b=(\Lambda ^b,\overrightarrow{\nu }^b)$ on $\Sigma ^b$ , which we will call the blow-up of $\overrightarrow{\lambda }=(\Lambda,\overrightarrow{\nu })$ .

Each of the measured laminations $\lambda ^b_i=(\Lambda ^b,\nu ^b_i)$ then induces a continuous everywhere-defined path pseudometric $d_{\lambda ^b_i}$ on $\widetilde{\Sigma ^b}$ .

Definition 3.4. The ${\mathbb{R}}^n$ -tree $T(\overrightarrow{\lambda })$ dual to $\overrightarrow{\lambda }$ is defined as the $\mathbb{R}$ -tree $T(\lambda )\;:\!=\;\Sigma ^b/d_\lambda ^b$ dual to the measured lamination $\lambda$ , endowed with the ${\mathbb{R}}^n$ -pseudometric given by the $n$ -tuple

\begin{align*}\overrightarrow {d}_\lambda \;:\!=\;(d_{\lambda _i})_{i=1,\dots,n}\end{align*}

of quotient pseudometrics $d_{\lambda _i}$ induced by $d_{\lambda ^b_i}$ .

It inherits an action of $\Gamma$ preserving $\overrightarrow{d}_\lambda$ , that is preserving each pseudometric $d_{\lambda _i}$ .

Proof. The crucial fact is that, by Proposition 3.1, the infima involved in the definition of the pseudometrics $d_{\lambda _i}$ are in fact all realized simultaneously for a same path $c$ in $\widetilde{\Sigma ^b}$ , the $m^b$ -geodesic. In particular, $c$ is a minimizing curve for $d_{\lambda ^b}$ and its projection on $T(\lambda )$ is a minimizing path for $\overrightarrow{d}_\lambda$ . This ensures that $d_{\lambda }=\sum _i d_{\lambda _i}$ on $T(\lambda )$ .

The separation of $\overrightarrow{d}_\lambda$ on $T(\lambda )$ follows: if $\overrightarrow{d}_\lambda (x,y)=0$ then $d_{\lambda }(x,y)=\sum _i d_{\lambda _i}(x,y)=0$ , hence $x=y$ .

4.1. Proof of Theorem 1.1

Theorem 1.1 is classical when $n=1$ and follows from the work of Thurston. Thurston furthermore proved [Reference Fathi, Laudenbach and Poénaru10, Expose 8] that $\mathcal{ML}$ is homeomorphic to ${\mathbb{R}}^{6g-6}$ , that it is projectivization ${\mathbb{P}}{\mathcal{ML}}$ is homeomorphic to $\mathbb S^{6g-7}$ , and that the resulting compactification is homeomorphic to a closed ball [Reference Martelli15, 8.3.13]. Using the latter fact, we can prove the following lemma, which ensures that given $L$ in $\mathcal{ML}$ , one can choose a sequence in $\mathcal{T}$ converging to $[L]$ with a fixed scale sequence. It will be crucial in the proof of Theorem 1.1.

Proof. Given a finite generating set $S\subset \Gamma$ and an isometric action $\rho$ of $\Gamma$ on a metric space $X$ , the minimal displacement of $\rho$ with respect to the generating set $S$ is defined by

\begin{align*}\lambda _S(\rho )\;:\!=\;\inf _{x\in X}\sqrt {\sum _s d(x,\rho (s)x)^2}\;.\end{align*}

This defines a nonzero proper and continuous function $\lambda _S \colon \mathcal{T} \to{\mathbb{R}}_+$ (this is for example proven—in a much more general context—in [Reference Parreau24, Prop. 25]).

We may suppose that $L$ is nonzero (otherwise we can take a constant sequence $\rho _k$ ). Let $T$ be a $\mathbb{R}$ -tree with minimal $\Gamma$ -action, with length function $L$ . Let $D$ be the minimal displacement of $\Gamma$ on $T$ with respect to the generating set $S$ .

Fix a point $[\rho _0]$ in the Teichmüller space $\mathcal{T}$ . As the compactification $\overline{\mathcal{T}}$ of $\mathcal{T}$ is a closed ball, there exists a path $r(t)$ , $t\in [0,1]$ from $[\rho _0]$ to $[L]$ in $\overline{\mathcal{T}}$ such that $r(t)$ belongs to $\mathcal{T}$ for $t\in [0,1[$ . Let $\rho _t$ be a representation with $[\rho _t]=r(t)$ . Then, as the map

\begin{align*}\begin {array}{c@{\quad}c@{\quad}c} [0,1[&\to &{\mathbb {R}}_+\\[5pt] t&\mapsto &\lambda _S(r(t)) \end {array}\end{align*}

is continuous and diverges as $t$ goes to $1$ , there exists an increasing sequence $(t_k)_{k\geq K}$ in $[0,1[$ with limit $1$ such that $\lambda _S(r(t_k))=D\lambda _k$ . Since $[\rho _k]=r(t_k)$ converges to $[L]$ in $\overline{\mathcal{T}}$ , we have that $\frac{1}{\lambda _S(\rho _k)} L_{\rho _k}$ converges to $sL$ for some $s\in{\mathbb{R}}_+$ . Taking the asymptotic cone of this sequence (see for example [Reference Parreau25]), we also have that $sL$ is the length function of an action of $\Gamma$ on a real tree $T_\omega$ with minimal displacement $1$ with respect to $S$ .Footnote 4 Let $T'\subset T_\omega$ be the minimal invariant subtree. As $T'$ is a convex subset of $T_\omega$ , the minimal displacement of $\Gamma$ in $T'$ is the same as in $T_\omega$ . By length rigidity of actions on minimal trees with length function in $\mathcal{ML}$ , the trees $T$ and $\frac{1}{s}T'$ are equivariantly isometric, hence have same minimal displacement $s=\frac{1}{D}$ . So we have that $\frac{1}{\lambda _k}L_{\rho _k} \to L$ as wanted.

We now have the ingredients needed to prove Theorem 1.1, which we recall for the reader’s convenience:

Proof. It is easy to see that $\partial ^{\textrm{WL}}(\mathcal{T}^n)\subset{\mathbb{P}}(({\mathcal{ML}})^n)$ . Let indeed $\rho _k=(\rho _{k,i})_{i=1,\ldots,n}$ , $k\in{\mathbb{N}}$ , be a sequence in ${\textrm{Hom}}(\Gamma,\textrm{PSL}(2,{\mathbb{R}})^n)$ , which we identify with ${\textrm{Hom}}(\Gamma,\textrm{PSL}(2,{\mathbb{R}}))^n$ . We write a nonzero function $\overrightarrow{L}\colon \Gamma \to{\mathbb{R}}^n$ as $\overrightarrow{L}=(L_i)_{i=1,\ldots,n}$ with $L_i\colon \Gamma \to{\mathbb{R}}$ . The sequence of conjugacy classes $[\rho _k]$ converges to the homothety class $[\overrightarrow{L}]$ in the Weyl chamber length compactification if and only if there exists a sequence of positive real numbers $\lambda _k\to \infty$ (scale sequence) such that the renormalized Weyl chamber length function $\frac{1}{\lambda _k}\overrightarrow{L}_{\rho _k}$ converges to $\overrightarrow{L}$ , that is, if $\frac{1}{\lambda _k}L_{\rho _{k,i}}$ converges to $L_i$ for all $i=1,\ldots,n$ [Reference Parreau25, Theorem 5.6]. Then either $L_i=0$ or $[\rho _{k,i}]$ converges to $[L_i]$ in the length compactification $\overline{\mathcal{T}}$ of $\mathcal{T}$ . As $\partial \mathcal{T}={\mathbb{P}}({\mathcal{ML}})$ , we have that each $L_i$ belongs to $\mathcal{ML}$ , hence $[\overrightarrow{L}]$ belongs to ${\mathbb{P}}(({\mathcal{ML}})^n)$ .

The converse implication is a consequence of Lemma 4.1. Let $\overrightarrow{L}=(L_i)_{i=1,\ldots,n}\in ({\mathcal{ML}})^n$ . Using Lemma 4.1 with scale sequence $\lambda _k=k$ , we can construct for each $i=1,\ldots,n$ a sequence of maximal representations $\rho _{k,i}:\Gamma \to \textrm{PSL}(2,{\mathbb{R}})$ , $k\in{\mathbb{N}}$ , whose renormalized length function $\frac{1}{k}L_{\rho _{k,i}}$ converges to the length function $L_i$ in $({\mathbb{R}}_+)^\Gamma$ as $k\to \infty$ . We now consider for $k\in{\mathbb{N}}$ the product representation $\rho _k\;:\! =\;(\rho _{k,i})_{i=1,\ldots,n}\colon \Gamma \to \textrm{PSL}(2,{\mathbb{R}})^n$ , which, being a product of maximal representations, is a maximal representation. Then $\frac{1}{k}\overrightarrow{L}_{\rho _k}=(\frac{1}{k}L_{\rho _{k,i}})_{i=1,\ldots,n}$ converges to $(L_i)_{i=1,\ldots,n}=\overrightarrow{L}$ as $k\to \infty$ . Hence, $[\overrightarrow{L}]$ belongs to $\partial ^{\textrm{WL}}(\mathcal{T}^n)$ .

As a result, we deduce that ${\mathbb{P}}(({\mathcal{ML}})^n)\subset{\mathbb{P}}(({\mathbb{R}}_+^\Gamma )^n)$ is a sphere of dimension $n(6g-6)-1$ , being the topological join of $n$ spheres:

4.2. Applications to other length compactifications

As an application of Proposition 4.2, we can also understand various other compactifications of $\Xi _{\text{Max}}(\Gamma,\textrm{PSL}(2,{\mathbb{R}})^n)$ :

In particular, the boundary of $\mathcal{T}^n$ in the $\ell ^1$ -length compactification is the projectivization ${\mathbb{P}}(\sum _{i=1}^n{\mathcal{ML}})$ of the space of geodesic currents that can be decomposed as the sum of $n$ measured laminations.

5.1. Flat structures

We will consider flat structures on a punctured finite type surfaces, namely the complement of a finite set of marked points considered as punctures in a compact topological surfaces. When dealing with mixed structures, the finite type surfaces will typically be obtained from geodesic subsurfaces of the original surface $\Sigma$ by collapsing each boundary components to a cusp point.

With a slight abuse of notation we denote by $\overrightarrow{\textrm{Flat}}(\Sigma )$ the moduli space of half-translation structures on $\Sigma$ , where two such structures are identified if they are isotopic. Note that the flat structures we consider here are directed (i.e.,, with a preferred vertical direction).

Remark 5.2. Duchin−Leiniger−Rafi [Reference Duchin, Leininger and Rafi9], as well as Ouyang−Tamburelli [Reference Ouyang and Tamburelli21] consider, instead, the space of flat structures on $\Sigma$ , where they identify isometric marked structures. As a result, we have fibrations

\begin{align*}\overrightarrow {\textrm {Flat}}(\Sigma )\to {\textrm {Flat}}(\Sigma )\to \mathcal {T}(\Sigma ).\end{align*}

The fiber of $\overrightarrow{\textrm{Flat}}(\Sigma )\to{\textrm{Flat}}(\Sigma )$ is the circle $\mathbb S^1$ , which acts on a half-translation surface by rotation. Since our structures are marked, and any half-translation structure induces a conformal structure on the surface, the space $\overrightarrow{\textrm{Flat}}(\Sigma )$ fibers over the Teichmüller space. We will never need this fact, but it is well known that the fiber over $X$ in this fibration identifies with the space of holomorphic quadratic differentials over the Riemann surface $X$ .

Let $K$ be a flat structure on $\Sigma$ , and let $\mathcal{F}_1$ , $\mathcal{F}_2$ be the vertical and horizontal measured foliations of $K$ . This is a pair of transverse measured foliations. For $i=1,2$ , we denote by

\begin{align*}\ell _{{K},i}(c)=i(\mathcal {F}_i,c)=\int _c|dx_i|\end{align*}

and by $d_{{K},i}\;:\!=\;d_{\mathcal{F}_i}$ the associated pseudometric on $\widetilde{\Sigma }$ This defines the natural ${\mathbb{R}}^2$ -metric

\begin{align*}\overrightarrow {d}_{K}=(d_{{K},i})_i\end{align*}

on $\widetilde{\Sigma }$ . We denote $L_{{K},i}\;:\!=\;I_{\mathcal{F}_i}\colon \Gamma \to{\mathbb{R}}$ the corresponding length function. Let $T(\mathcal{F}_i)\;:\!=\;\widetilde{\Sigma }/d_{{K},i}$ be the dual tree of $\mathcal{F}_i$ , and $p_i\colon \widetilde{\Sigma }\to T(\mathcal{F}_i)$ the corresponding projection.

The universal cover $\widetilde{\Sigma }$ of $\Sigma$ is a CAT(0) metric space for the flat metric associated to $K$ , which we will denote by $d_{{\textrm{CAT}}(0)}$ . Note that $d_{{\textrm{CAT}}(0)}$ is not in general equal to the metric $\|\overrightarrow{d}_{K}\|_2$ induced from the ${\mathbb{R}}^2$ -metric $\overrightarrow{d}$ by taking the $\ell ^2$ -norm, nevertheless it is the associated length metric. We denote $\widetilde{\Sigma }^c$ the completion of $\widetilde{\Sigma }$ with respect to $d_{{\textrm{CAT}}(0)}$ , which consists in adding one fixed point $x_{{\widetilde{c}}}$ for each parabolic subgroup $\Gamma _{{\widetilde{c}}}$ of $\Gamma =\pi _1(\Sigma )$ (corresponding to lifts $\widetilde{c}$ of punctures $c$ of $\Sigma$ ), see for example [Reference Morzadec18, Lemma 7.2]. The flat structure extends on $\widetilde{\Sigma }^c$ . It is easy to see that $\mathcal{F}_i$ , $d_{{K},i}$ , and $p_i$ extend naturally to the completion $\widetilde{\Sigma }^c$ of $\widetilde{\Sigma }$ .

We define the $\ell ^1$ -length metric on $\widetilde{\Sigma }$ , as the Finsler metric $d_{K}$ induced by the