Geodesic
Weyl chamber length compactification of the $\textrm{PSL}(2,{\mathbb{R}})\times \textrm{PSL}(2,{\mathbb{R}})$ maximal character variety
Glasgow mathematical journal, Tome 67 (2025) no. 1, pp. 11-33

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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.
DOI : 10.1017/S0017089524000156
Mots-clés : compactifications of character varieties, surface groups, maximal representations, products of trees 
Burger, Marc; Iozzi, Alessandra; Parreau, Anne; Pozzetti, Maria Beatrice. Weyl chamber length compactification of the $\textrm{PSL}(2,{\mathbb{R}})\times \textrm{PSL}(2,{\mathbb{R}})$ maximal character variety. Glasgow mathematical journal, Tome 67 (2025) no. 1, pp. 11-33. doi: 10.1017/S0017089524000156
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     author = {Burger, Marc and Iozzi, Alessandra and Parreau, Anne and Pozzetti, Maria Beatrice},
     title = {Weyl chamber length compactification of the $\textrm{PSL}(2,{\mathbb{R}})\times \textrm{PSL}(2,{\mathbb{R}})$ maximal character variety},
     journal = {Glasgow mathematical journal},
     pages = {11--33},
     year = {2025},
     volume = {67},
     number = {1},
     doi = {10.1017/S0017089524000156},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089524000156/}
}
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%A Parreau, Anne
%A Pozzetti, Maria Beatrice
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