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

TL;DR

This paper explores the boundary structure of the space of maximal surface group representations into ${ m PSL}(2,b R)^n$, identifying it with measured lamination spaces and providing geometric interpretations for the case n=2.

Contribution

It introduces a vectorial length compactification for maximal representations and characterizes its boundary with geometric and algebraic structures, including dual tree-graded spaces.

Findings

Boundary identified with sphere of measured lamination n-tuples

Geometric interpretation of boundary as classes of mixed structures for n=2

Universal dual space with tree-graded structure and $b R_+^2$-metric

Abstract

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in ${\rm PSL}(2,\mathbb R)^n$. We identify the boundary with the sphere $\mathbb P((\mathcal{ML})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of $\mathbb R^2$-mixed structures on $\Sigma$. We associate to such a structure a dual tree-graded space endowed with an $\mathbb R_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb R$-trees with the given length spectrum.