Weyl chamber length compactification of the PSL(2, ℝ) × PSL(2, ℝ) 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 ∑ 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 ∑. In the case n = 2, we give a geometric interpretation of the boundary as the space of homothety classes of ℝ2-mixed structures on ∑. We associate to such a structure a dual tree-graded space endowed with an ℝ2+-valued metric, which we show to be universal with respect to actions on products of two ℝ-trees with the given length spectrum.