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 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.

Geometry of the word problem for 3-manifold groups

“…Proofs of the results in this section and more detailed background on autostackability are in [8,9,15]. Let G = A be an autostackable group, with spanning tree T in Γ A (G) and flow function Φ : E → P .…”

“…In [7] the first two authors show that if G is a stackable group whose flow function bound is K, then G is finitely presented with relators given by the labels of loops in Γ A (G) of length at most K + 1 (namely the relations φ(nf(g), a) = G a). Although it is unknown if autostackability is invariant under changes in finite generating sets, we note that it is straightforward to show that if G is autostackable with generating set A and A ⊆ B, then G is autostackable with the generators B using the same set of normal forms (see [15,Proposition 4.3] for complete details).…”

“…Not every stackable group has decidable word problem; Hermiller and Martínez-Pérez [15] show that there exist groups with a bounded flow function but unsolvable word problem. A group with a bounded flow function Φ whose graph is a recursive (i.e., decidable) language has a word problem solution using the automaton that recognizes graph (Φ) [8].…”

“…The class of autostackable groups contains all groups with a finite convergent rewriting system or an asynchronously automatic structure with prefix-closed (unique) normal forms [8]. Beyond these examples, autostackable groups include some groups that do not have homological type F P 3 [9, Corollary 4.2] and some groups whose Dehn function is nonelementary primitive recursive; in particular, Hermiller and Martínez-Pérez show in [15] that the Baumslag-Gersten group is autostackable.…”

“…For groups with finite convergent rewriting systems, closure for HNNextensions in which one of the associated subgroups equals the base group and the other has finite index in the base group is given in [14]. Closure for stackable groups in the special case of an HNN extension under significantly relaxed assumptions (and using left cosets instead of right) are given by the second author and Martínez-Pérez in [15]. They also prove a closure result for HNN extensions of autostackable groups, with a requirement of further technical assumptions.…”