Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Atoroidal dynamics of subgroups of Out(F_N)

Matt Clay, Caglar Uyanik

Abstract

We show that for any subgroup $H$ of Out($F_N$), either $H$ contains an atoroidal element or a finite index subgroup $H'$ of $H$ fixes a nontrivial conjugacy class in $F_N$. This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel-Mosher's subgroup theorem for Out($F_N$) in the setting of irreducible elements.

Atoroidal dynamics of subgroups of Out(F_N)

Abstract

We show that for any subgroup of Out(), either contains an atoroidal element or a finite index subgroup of fixes a nontrivial conjugacy class in . This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel-Mosher's subgroup theorem for Out() in the setting of irreducible elements.

Paper Structure

This paper contains 15 sections, 26 theorems, 47 equations, 2 figures.

Table of Contents

  1. Outer automorphisms and train tracks
  2. Graphs, maps and markings
  3. Free factor systems and geometric realizations
  4. Relative outer automorphisms
  5. Train tracks and CTs
  6. Geodesic currents
  7. Pushing past multi-edge extensions
  8. Dynamics on single-edge extensions
  9. Almost atoroidal elements
  10. North-south dynamics for atoroidal elements
  11. Completely split goodness of paths and currents
  12. Incorporating north-south dynamics from lower stratum
  13. Generalized north-south dynamics for almost atoroidal elements
  14. Pushing past single-edge extensions
  15. Proof of the subgroup alternative

Key Result

Theorem A

Let $\mathcal{H}$ be a subgroup of $\mathop{\mathrm{Out}}\nolimits(F_{N})$ where $N \geq 3$. Either $\mathcal{H}$ contains an atoroidal element or there exists a finite index subgroup $\mathcal{H}'$ of $\mathcal{H}$, and a nontrivial element $g\in F_{N}$ such that $\mathcal{H}'[g]=[g]$.

Figures (2)

  • Figure 1: The set-up of neighborhoods in Theorem \ref{['th:gns']}. For $n \geq M$, the element $\varphi^{n}$ sends the complement of $\widehat{V}_{-}$ into $U_{+}$; the element $\varphi^{-n}$ sends the complement of $\widehat{V}_{+}$ into $U_{-}$.
  • Figure 2: The set-up of neighborhoods in $\mathbb{P} {\rm Curr}(F)$ for Proposition \ref{['prop:atoroidal']}.

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • proof
  • Theorem \oldthetheorem: HMIntro
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem: FH
  • Lemma \oldthetheorem: Ka2
  • Theorem \oldthetheorem
  • ...and 37 more