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

TL;DR

This paper establishes the equivalence between primitive stability and a generalized form of Q-conditions for representations of the rank two free group into hyperbolic d-space, extending known results to higher dimensions.

Contribution

It generalizes the equivalence of primitive stability and Q-conditions to hyperbolic spaces of dimension three and higher, under specific assumptions, and extends the bounded intersection property to these settings.

Findings

Primitive stability and Q-conditions are equivalent for F_2 representations in hyperbolic d-space.

The results apply to all W_3-extensible representations.

The bounded intersection property is extended to higher dimensions.

Abstract

The two largest known domains of discontinuity for the action of Out(F_2) on the PSL(2,C)-character variety of F_2 - 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 F_2 into the isometry group of hyperbolic d-space for d >= 3, under some assumptions. In particular, these assumptions are satisfied by all W_3-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.