Automaticity for graphs of groups

Susan Hermiller, Derek F. Holt, Sarah Rees and Tim Susse


Automatic groups, automatic coset system, graph of groups, relatively hyperbolic group, 3-manifold group, Artin group


Published in Geometry, Groups and Dynamics 15 (2021) 503--551


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.

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