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Morse and stable subgroups via the coset intersection complex

Tomohiro Fukaya, Haoyang He, Eduardo Martínez-Pedroza, Takumi Matsuka

Abstract

In this note, we study the equivalence of Morse and stable subgroups in the framework of the coset intersection complex. Under certain conditions on a coset intersection complex of a group, we prove that infinite-index Morse subgroups are stable. Our main theorem recovers results in the literature on right-angled Artin groups and graph products. As an application, we show that for the genus-two handlebody group, any infinite-index Morse subgroup is stable.

Morse and stable subgroups via the coset intersection complex

Abstract

In this note, we study the equivalence of Morse and stable subgroups in the framework of the coset intersection complex. Under certain conditions on a coset intersection complex of a group, we prove that infinite-index Morse subgroups are stable. Our main theorem recovers results in the literature on right-angled Artin groups and graph products. As an application, we show that for the genus-two handlebody group, any infinite-index Morse subgroup is stable.

Paper Structure

This paper contains 6 sections, 18 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem
  3. Proof of the corollary
  4. Graph products
  5. Groups that acts cocompactly and acylindrically on trees
  6. The genus-two handlebody group

Key Result

Theorem 1.2

Let $G$ be a finitely generated group. Let $H$ be an infinite subgroup of $G$. Then $H$ is stable if and only if $H$ is Morse and hyperbolic.

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2: MR3956891
  • Theorem 1.3
  • Definition 1.4
  • Theorem \ref{thm_Mstablevf}
  • Corollary 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: MR3956891
  • ...and 22 more