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This is a Quantitative Study study.

October 2, 2025Open Access

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

Key Points

The research establishes that primitive stability is equivalent to the generalized Q-conditions for F_2 representations.
Key findings suggest that all W_3-extensible representations satisfy the established assumptions for this equivalence.
Generalizations of previous results on bounded intersection property are extended to higher dimensions, revealing new insights.
The equality of the two domains of discontinuity for Out(F_2) actions was independently established by Lee-Xu and Series.

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.

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

Key Points

Abstract

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