Weyl chamber length compactification of the maximal character variety

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group Γ of a closed hyperbolic surface Σ in PSL(2,R)ⁿ . We identify the boundary with the sphere P((ML)ⁿ) , where ML is the space of measured geodesic laminations on Σ . In the case $n=2$ , we give a geometric interpretation of the boundary as the space of homothety classes of R² -mixed structures on Σ . We associate to such a structure a dual tree-graded space endowed with an R₊² -valued metric, which we show to be universal with respect to actions on products of two R -trees with the given length spectrum.