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Cohomological and quasi-isometric diversity of groups with property $R_\infty$

Karel Dekimpe, Paula M. Lins de Araujo, Yuri Santos Rego

Abstract

How rich is the collection of groups with a given prominent property? In this work we approach this question for property~$R_\infty$, which says that every automorphism $\varphi$ of a given group has infinitely many orbits under the $\varphi$-twisted conjugation action $(g,x) \mapsto gx\varphi(g)^{-1}$. Generalising the soluble groups of Herbert Abels to a large family over many integral domains, we prove that most such groups have property~$R_\infty$ drawing from a classical result of Levchuk and a swift observation by Jabara. Within the broad programme of cataloguing finitely generated groups up to quasi-isometry, our groups can then be separated by finiteness properties and cohomological dimension whilst having~$R_\infty$. Abandoning finite presentability, we establish that property~$R_\infty$ is very abundant in a strong sense: there are uncountably many finitely generated groups (which can all be chosen to be amenable or non-amenable) that have~$R_\infty$ and are pairwise not quasi-isometric. The proofs vary in flavour. On the amenable side we use carefully constructed quotients of Abels' groups and a general strategy for quasi-isometric diversity established by Minasyan, Osin, and Witzel. For the non-amenable constructions we rely on modifications of Leary's type $\mathtt{FP}$ groups, further cohomological arguments, and recent powerful criteria for~$R_\infty$ due to Iveson, Martino, Sgobbi, Wong, and Fournier-Facio.

Cohomological and quasi-isometric diversity of groups with property $R_\infty$

Abstract

How rich is the collection of groups with a given prominent property? In this work we approach this question for property~, which says that every automorphism of a given group has infinitely many orbits under the -twisted conjugation action . Generalising the soluble groups of Herbert Abels to a large family over many integral domains, we prove that most such groups have property~ drawing from a classical result of Levchuk and a swift observation by Jabara. Within the broad programme of cataloguing finitely generated groups up to quasi-isometry, our groups can then be separated by finiteness properties and cohomological dimension whilst having~. Abandoning finite presentability, we establish that property~ is very abundant in a strong sense: there are uncountably many finitely generated groups (which can all be chosen to be amenable or non-amenable) that have~ and are pairwise not quasi-isometric. The proofs vary in flavour. On the amenable side we use carefully constructed quotients of Abels' groups and a general strategy for quasi-isometric diversity established by Minasyan, Osin, and Witzel. For the non-amenable constructions we rely on modifications of Leary's type groups, further cohomological arguments, and recent powerful criteria for~ due to Iveson, Martino, Sgobbi, Wong, and Fournier-Facio.
Paper Structure (24 sections, 40 theorems, 53 equations)

This paper contains 24 sections, 40 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Motivation, context and open questions
  3. Outline of the paper
  4. Background
  5. Setup, notation, and auxiliary results
  6. Finiteness properties, dimension, and quasi-isometries
  7. The groups SIn(R)
  8. Useful properties of PBn(R) and SIn(R)
  9. The group PBn(R) has R-infinity
  10. Automorphisms of upper triangular matrix groups
  11. Automorphisms of PBn(R)
  12. Proof of \ref{['mainthm']} for PBn(R)
  13. The general case
  14. Automorphisms of SI(R)
  15. Proof of Theorem \ref{['mainthm']} for SI(R)
  16. ...and 9 more sections

Key Result

Theorem 1.1

The groups $\mathbf{S}^I_n(R)$ can be separated by finiteness properties and cohomological dimension while having property $R_\infty$ and being finitely presented. That is to say, for every $n\geq 4$ there exists an infinite subset $D_n \subseteq \mathbb{N}$ such that, given $d \in D_n \cup \{\infty

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Conjecture 1.7
  • Lemma 2.1: E.g., FelshtynTroitskyCrelle
  • Lemma 2.2: E.g., DacibergWongCrelle
  • Lemma 2.3
  • Lemma 2.4: Jabara's trick Jabara
  • ...and 35 more