1 results
Weyl chamber length compactification of the
$\textrm{PSL}(2,{\mathbb{R}})\times \textrm{PSL}(2,{\mathbb{R}})$ maximal character variety
- Part of
-
- Journal:
- Glasgow Mathematical Journal , First View
- Published online by Cambridge University Press:
- 27 September 2024, pp. 1-23
-
- Article
-
- You have access
- HTML
- Export citation
-
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.
