Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Acylindrically hyperbolic groups and counting problems

Inhyeok Choi

TL;DR

This work proves that Morse elements are generic in any acylindrically hyperbolic group, establishing that in large word balls, almost all elements exhibit Morse behavior. The authors connect Morse genericity to the presence of a WPD loxodromic action on a hyperbolic space, deriving a density-1 result for Morse elements and extending the conclusion to applications in $\mathrm{Out}(F_N)$, where fully irreducible ageometric triangular elements become generic in $\mathrm{Out}(F_N)$ with respect to any finite generating set. The approach diverges from random-walk results by focusing on counting in ball metrics, and it leverages a variant of geometric separation, projection-alignment techniques, and Behrstock-type inequalities to bound non-Morse contributions. The paper also discusses extensions beyond hyperbolic spaces (e.g., CAT(0) rank-1 settings), related open questions about exponential genericity across word metrics, and contrasts with similar results in mapping class groups and related groups. Overall, the method provides a robust counting framework for Morse elements via WPD-induced contraction, with broad implications for the structure of automorphism groups of free groups and other acylindrically hyperbolic groups.

Abstract

We show that Morse elements are generic in acylindrically hyperbolic groups. As an application, we observe that fully irreducible outer automorphisms are generic in the outer automorphism group of a finite-rank free group.

Acylindrically hyperbolic groups and counting problems

TL;DR

This work proves that Morse elements are generic in any acylindrically hyperbolic group, establishing that in large word balls, almost all elements exhibit Morse behavior. The authors connect Morse genericity to the presence of a WPD loxodromic action on a hyperbolic space, deriving a density-1 result for Morse elements and extending the conclusion to applications in , where fully irreducible ageometric triangular elements become generic in with respect to any finite generating set. The approach diverges from random-walk results by focusing on counting in ball metrics, and it leverages a variant of geometric separation, projection-alignment techniques, and Behrstock-type inequalities to bound non-Morse contributions. The paper also discusses extensions beyond hyperbolic spaces (e.g., CAT(0) rank-1 settings), related open questions about exponential genericity across word metrics, and contrasts with similar results in mapping class groups and related groups. Overall, the method provides a robust counting framework for Morse elements via WPD-induced contraction, with broad implications for the structure of automorphism groups of free groups and other acylindrically hyperbolic groups.

Abstract

We show that Morse elements are generic in acylindrically hyperbolic groups. As an application, we observe that fully irreducible outer automorphisms are generic in the outer automorphism group of a finite-rank free group.
Paper Structure (9 sections, 1 theorem, 85 equations, 2 figures)

This paper contains 9 sections, 1 theorem, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Comparison with other groups
  3. Another side of the story: random walks
  4. Beyond hyperbolic spaces
  5. Open questions
  6. Plans
  7. Preliminaries
  8. WPD property and contraction
  9. Proof of Theorem \ref{['thm:mainWPD']}

Key Result

Theorem A

Let $G$ be an acylindrically hyperbolic group. Then for any finite generating set $S$ of $G$, we have

Figures (2)

  • Figure 1: Schematics for $f(D, M)$ in Subsection \ref{['subsection:compare']}
  • Figure 2: Properties of the four actions and the density estimates

Theorems & Definitions (13)

  • Theorem A
  • proof
  • Definition 2.5
  • Definition 2.7
  • proof
  • proof
  • proof : Proof of Lemma \ref{['lem:bad']}
  • Claim 4.4
  • proof : Proof of Claim \ref{['claim:maxRed']}
  • proof
  • ...and 3 more