Homology and closure properties of autostackable groups

TL;DR

This paper explores the properties of autostackable groups, showing they include some groups not of type $FP_3$ and are closed under graph products and extensions, expanding understanding of their algebraic and topological features.

Contribution

It demonstrates that autostackable groups can lack certain homological finiteness properties and are closed under specific group operations, broadening the class's known structural characteristics.

Findings

Includes groups not of type $FP_3$

Closed under graph products

Closed under extensions

Abstract

Autostackability for finitely presented groups is a topological property of the Cayley graph combined with formal language theoretic restrictions, that implies solvability of the word problem. The class of autostackable groups is known to include all asynchronously automatic groups with respect to a prefix-closed normal form set, and all groups admitting finite complete rewriting systems. Although groups in the latter two classes all satisfy the homological finiteness condition $FP_\infty$, we show that the class of autostackable groups includes a group that is not of type $FP_3$. We also show that the class of autostackable groups is closed under graph products and extensions.