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

Balthazar Fléchelles

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.

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

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.

Paper Structure

This paper contains 28 sections, 22 theorems, 60 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Q-conditions
  3. Primitive displacement
  4. Primitive stability
  5. Known implications
  6. Results
  7. Automorphisms and primitive elements
  8. Primitive elements, bases and automorphisms
  9. (Outer) automorphism groups
  10. Conjugated actions on character varieties
  11. The Farey tessellation
  12. Christoffel words
  13. The half-length property
  14. Reminders on hyperbolic geometry
  15. Definition
  16. ...and 13 more sections

Key Result

theorem 1

Let $\rho: F_2\to\mathop{\mathrm{Isom}}\nolimits(\mathbb{H}^{d})$ be a Coxeter extensible representation. There exists $\lambda >0$ such that $\rho$ is primitive stable if and only if it satisfies the $Q_\lambda$-conditions.

Figures (7)

  • Figure 1: (Poincaré model of $\mathbb{H}^{d}$) The axes of the pair $(AB,B)$ are in the same configuration as those of the pair $(A,B)$, but $\ell_{\mathbb{H}^{d}}(AB) < \ell_{\mathbb{H}^{d}}(A)$. Iterating gives a contradiction with the $Q$-conditions
  • Figure 2: The right-angled hexagon $\mathcal{H}(U,V)$ decorated by its orthogonal hyperplanes. A plane of the same color as the axis of an isometry is orthogonal to this axis. The black lines are contained in the intersection of the two hyperplanes around it
  • Figure 3: In these two configurations, we can guarantee that some hyperplanes do not intersect
  • Figure 4: Let $P$ denote the polar of $\mathop{\mathrm{\perp}}\nolimits(\Delta,\mathcal{Q})$ in $B^2$. The other points are defined on the diagram. In the right triangle $ONM$, Pythagoras's theorem gives $\lVert OM\lVert = \sqrt{1-(d/2)^2}$. Since the triangles $ONM$ and $ONP$ are similar, we have $\lVert OP\lVert = 1/\sqrt{1-(d/2)^2}$
  • Figure 5: (Poincaré model of $\mathbb{H}^{d}$) If $hg^+ = g^-$, then if we choose $o = o_n$ and the corresponding $o'_n$ so that $g^nho'_n = o_n$, we see that $\ell_{\mathbb{H}^{d}}(g^nh)\to 0$
  • ...and 2 more figures

Theorems & Definitions (72)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • theorem 1: see Theorem \ref{['thm:PSisPDisBQ']}
  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • ...and 62 more