HNN extensions and stackable groups

TL;DR

This paper explores the properties of stackable groups, introduces new characterizations, and demonstrates that HNN extensions of such groups remain stackable, revealing diverse Dehn functions and examples with unsolvable word problems.

Contribution

It provides two new characterizations of stackable groups and proves that HNN extensions preserve stackability, expanding understanding of group structures and their algorithmic properties.

Findings

HNN extensions of stackable groups are also stackable

Existence of stackable groups with unsolvable word problem

Diverse Dehn functions admitted by stackable and autostackable groups

Abstract

Stackability for finitely presented groups consists of a dynamical system that iteratively moves paths into a maximal tree in the Cayley graph. Combining with formal language theoretic restrictions yields auto- or algorithmic stackability, which implies solvability of the word problem. In this paper we give two new characterizations of the stackable property for groups, and use these to show that every HNN extension of a stackable group is stackable. We apply this to exhibit a wide range of Dehn functions that are admitted by stackable and autostackable groups, as well as an example of a stackable group with unsolvable word problem. We use similar methods to show that there exist finitely presented metabelian groups that are non-constructible but admit an autostackable structure.