# Automaticity for graphs of groups

## TL;DR

This paper develops automatic structures for complex groups like amalgamated products and HNN extensions, broadening the understanding of automaticity in fundamental groups of graphs of groups under geometric conditions.

## Contribution

It introduces methods to construct automatic structures for a wide class of groups, including Artin, Coxeter, and hyperbolic relative to abelian groups, under geometric conditions.

## Key findings

- Automatic structures constructed for amalgamated products and HNN extensions.
- Results apply to Artin groups of large type, Coxeter groups, and relatively hyperbolic groups.
- Closure properties established under geometric conditions.

## 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.