Infinitely presented simple groups separated by homological finiteness properties

TL;DR

The paper addresses constructing infinitely presented simple groups with controlled homological finiteness properties by leveraging Röver--Nekrashevych groups. The authors show how to transfer finiteness properties from a linear group $G\le\mathrm{GL}_n(\mathbb{Q})$ to a simple group via persistent/quasi-retract frameworks within $V_d(G)$ and its extensions, yielding new examples of simple groups of type $FP_n$ (including $FP_{\infty}$) that are not finitely presented, and establishing Dehn-function lower bounds inherited from linear subgroups. Central techniques include the Stein--Farley complex, Brown's criterion, and constructions that enforce finite abelianization while preserving finiteness properties, along with self-similar actions of affine groups and Baumslag--Solitar extensions to realize persistent retracts. The results significantly broaden the landscape of simple groups with prescribed homological finiteness, provide tools for obtaining fast-growing Dehn functions in simple groups, and raise new questions about embedding linear groups into simple Thompson-like groups.

Abstract

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $Γ$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type $\mathrm{FP}_{\infty}$ that is not finitely presented. Moreover we show that for every $n \in \mathbb{N}$ there is a simple group of type $\mathrm{FP}_n$ that is neither finitely presented nor of type $\mathrm{FP}_{n+1}$. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.

Theorems & Definitions (61)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.7
• Corollary 1.8
• Proposition 1.9
• Definition 2.1
• Definition 2.2