HNN extensions and stackable 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 theword problem. In this paperwe give two newcharacterizations of the stackable property for groups, and use these to show that every HNN extension of a stackable group over finitely generated subgroups 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 similarmethods to show that there exist finitely presented metabelian groups that are non-constructible but admit an autostackable structure.