Abstract
In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associated automatic subgroups, subject to further geometric conditions. These results are proved in the general context of fundamental groups of graphs of groups. The hypotheses of our closure results are satisfied in a variety of examples such as Artin groups of sufficiently large type, Coxeter groups, virtually abelian groups, and groups that are hyperbolic relative to virtually abelian subgroups.
Highlights
Closure properties for the classes of automatic and asynchronously automatic groups are known for a variety of group constructions
The special case of Corollary 4.10 in which the graph of groups arises from the JSJ decomposition of a 3-manifold yields Corollary C, which gives new automatic structures for fundamental groups of these 3-manifolds with respect to a Higgins language of normal forms
These fundamental groups were first shown to be automatic by Epstein et al [16, Thm. 12.4.7] and Shapiro [29], but the structure of the associated languages is not transparent from the proofs; the results of Dahmani and of Antolin and Ciobanu provide a shortlex automatic structure
Summary
Closure properties for the classes of automatic and asynchronously automatic groups are known for a variety of group constructions. These are the most general combination theorems of this article, building respectively strong asynchronous and strong synchronous coset systems for π1(G), given appropriate conditions of crossover and stabliity on vertex, edge subgroup pairs (and in the second case some further conditions). We apply this to derive Theorem B, our general closure result for graphs of groups that are strongly synchronously coset automatic. Given a group G hyperbolic relative to a set of subgroups, and a specified such subgroup H, the technical result Proposition 4.5 provides conditions under which we can find a strong synchronous automatic coset system for (G, H) that satisfies crossover and some other conditions we need. We prove assorted results on the strong synchronous coset automaticity of various types of amalgamated free products G1 ∗H G2 for which (G1, H) and (G2, H) are both strongly coset automatic; in particular Proposition 6.5 proves the strongly synchronous coset automaticity of an amalgamated product of a finitely generated abelian group and a group that is hyperbolic relative to a collection of abelian subgroups, where amalgamation is over one of those subgroups
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