Primitive stability and the Q-conditions for the rank two free group in hyperbolic d-space

2025/07/13 by Fléchelles, Balthazar
#14M35 #20E05 #20E36 #51M10 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2507.09771

Abstract

The two largest known domains of discontinuity for the action of Out(F₂) on the PSL(2,C)-character variety of F₂ - defined by Minsky's primitive stability, and Bowditch's Q-conditions - were proven to be equal independently by Lee-Xu and Series. We prove the equivalence between primitive stability and a generalization of the Q-conditions for representations of F₂ into the isometry group of hyperbolic d-space for d >= 3, under some assumptions. In particular, these assumptions are satisfied by all W₃-extensible representations. We also generalize Lee-Xu's and Series' results concerning the bounded intersection property to higher dimensions after extending their original definition to this setting.

Citations

• On some stable representations of hyperbolic groups
• Equivalence of primitive-stable and Bowditch actions of the free group of rank two on Gromov-hyperbolic spaces
• Primitive stability and the Bowditch conditions revisited
• Primitive stability and Bowditch's BQ-condition are equivalent
• The marked length spectrum of a projective manifold or orbifold
• On dynamics of Out(F_n) on PSL(2,C) characters
• Discreteness Criteria and the Hyperbolic Geometry of Palindroms
• Enumerating Palindromes and Primitives in Rank Two Free Groups
• Markoff triples and quasifuchsian groups

Related

• Equivalence of primitive-stable and Bowditch actions of the free group of rank two on Gromov-hyperbolic spaces
• Bowditch representations in Gromov-hyperbolic spaces : characterizations, dynamics of Out(\mathbbF₂) and recognition
• Primitive stability and the Bowditch conditions revisited
• On dynamics of the Mapping class group action on relative PSL(2,ℝ)-Character Varieties
• Primitive stability and Bowditch's BQ-condition are equivalent