Geometry of the word problem for 3-manifold groups

Abstract

We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup.Comment: 32 pages, 1 figur

Automaticity for graphs of groups

“…In this section we use Lemma 4.4 to show that a group that is hyperbolic relative to geodesically biautomatic subgroups is coset automatic relative to each peripheral subgroup with maximal crossover. We note that a similar but weaker SSCA result is shown in [8,Theorem 5.4].…”

“…Now we provide a description of the language for π 1 (G) that we use in our proof of Theorem A. This is a set of words representing normal forms provided by Higgins in [18], but modified to work with right rather than left cosets, and to provide words over generating sets rather than normal forms that are products of elements; see [8,Prop. 3.3] for more details.…”

“…Let L τ (e0) be a regular language of normal forms for G τ (e0) . It is shown in [8,Prop. 3.3] that the Higgins language for π 1 (G), with respect to L, L τ (e0) , τ (e 0 ), is a regular language; the same proof shows that the Higgins coset language L is also regular.…”

“…12.4.7] to be automatic, the normal forms for the automatic structure are difficult to determine from the construction in that proof. The proofs of Theorems A and 4.8 were partly inspired by the proof in [8] that all fundamental groups of closed 3-manifolds have the related property of autostackability, and as in that earlier proof, our proofs of those theorems use the set of Higgins normal forms described in Section 3.1. We now show that when the pieces of the JSJ decomposition are hyperbolic, the fundamental group of the 3-manifold is also automatic over those normal forms.…”

HNN extensions and stackable groups

“…An autostackable structure for a finitely generated group implies a finite presentation, a solution to the word problem, and a recursive algorithm for building van Kampen diagrams [6]. Moreover, in contrast to automatic groups, Brittenham and Hermiller together with Susse have shown that the class of autostackable groups includes all fundamental groups of 3-manifolds [9], with Holt they have shown autostackable examples of solvable groups that are not virtually nilpotent [7], and with Johnson they show that Stallings' non-F P 3 group [8] is autostackable. In analogy with the relationship between automatic and combable groups, removing the formal language theoretic restriction gives the stackable property for finitely generated groups, and stackability implies tame combability [5].…”