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

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 $\textrm{PSL}(2,{\mathbb{R}})^n$ . 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.