Limit cones of multi-Fuchsian representations

2025/12/04 by Danciger, Jeffrey, Guéritaud, François, Kassel, Fanny · 1 citation
Mathematics · #Advanced Operator Algebra Research #Boundary (topology) #Cone (formal languages) #Convex cone #Convex set #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Limit (mathematics) #Limit set #Regular polygon #Surface (topology)

paper · open access · doi:10.48550/arxiv.2512.04684

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/12/04 · openalex created_date 2025/12/06 · openalex updated_date 2026/07/28

Abstract

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into (PSL₂R)^d whose projections to PSL₂R are all convex cocompact. These multi-lengths define a convex cone in R^d_{≥ 0}, called the limit cone. When d=3, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone. We also give examples where the limit cone varies discontinuously with the representation.

Citations

• Flexibility and degeneracy around a theorem of Thurston
• Convex structures of the unit tangent spheres in Teichmüller space
• Deformations of Anosov subgroups: Limit cones and Growth indicators
• Morse actions of discrete groups on symmetric spaces: Local-to-global principle
• Infinitesimal Zariski closures of positive representations
• Ergodic optimization in dynamical systems
• Ergodic optimization of Birkhoff averages and Lyapunov exponents
• Lengthening deformations of singular hyperbolic tori
• The pressure metric for Anosov representations
• Higher Teichmüller Spaces: from SL(2,R) to other Lie groups
• Dual Teichmuller and lamination spaces
• Markoff triples and quasifuchsian groups
• Minimal stretch maps between hyperbolic surfaces

Cited by

• Flexibility and degeneracy around a theorem of Thurston

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