Weyl chamber length compactification of the \rm PSL(2,\mathbb R)×\rm PSL(2,\mathbb R) maximal character variety

2021/12/27 by Marc Burger, Burger, Marc, Alessandra Iozzi +5
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2112.13624

openalex publication_date 2021/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \rm PSL(2,\mathbb R)ⁿ. We identify the boundary with the sphere \mathbb 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 \mathbb R²-mixed structures on Σ. 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.

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