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Finiteness properties and quasi-isometry of group pairs

Kevin Li, Luis Jorge Sánchez Saldaña

Abstract

We show that the geometric and homological finiteness properties of group pairs are invariant under a suitable notion of quasi-isometry for group pairs.

Finiteness properties and quasi-isometry of group pairs

Abstract

We show that the geometric and homological finiteness properties of group pairs are invariant under a suitable notion of quasi-isometry for group pairs.
Paper Structure (7 sections, 24 theorems, 23 equations)

This paper contains 7 sections, 24 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Quasi-isometry of group pairs
  3. Finiteness properties of group pairs
  4. Quasi-isometry invariance
  5. Homological finiteness properties of group pairs
  6. Relative finite presentability
  7. Bredon finiteness properties for families

Key Result

Theorem 1.1

Let $G$ and $H$ be finitely generated groups and let $n\in \mathbb{N}$ with $n\ge 2$. Assume that $G$ is a quasi-retract of $H$. Then the following hold: In particular, if $G$ and $H$ are quasi-isometric, then $G$ is of type $\mathsf{FP}_n$ (resp. of type $\mathsf{F}_n$) if and only if $H$ is of type $\mathsf{FP}_n$ (resp. of type $\mathsf{F}_n$).

Theorems & Definitions (56)

  • Theorem 1.1: Alonso Alonso94
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Theorem \ref{['thm:FPn_QI']} and Corollary \ref{['cor:Fn_QI']}
  • Theorem 1.6: Corollary \ref{['cor:Bredon_QI']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: HMPSS
  • Lemma 2.4: HMPSS
  • ...and 46 more