An automata theoretic approach to the generalized word problem in graphs of groups

TL;DR

This paper presents an automata-theoretic proof demonstrating that certain graphs of groups maintain decidability of the generalized word problem, simplifying previous approaches and covering a broad class of groups including chordal graph groups.

Contribution

It provides a simpler automata-based proof for the preservation of decidability in graphs of groups, extending previous results to include more general classes such as chordal graph groups.

Findings

Automata theory can simplify proofs of group-theoretic properties.

Benign graphs of groups preserve decidability of the generalized word problem.

Includes classes like hyperbolic and polycyclic-by-finite groups.

Abstract

We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which so-called benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups in which edge groups are polycyclic-by-finite and vertex groups are either locally quasiconvex hyperbolic or polycyclic-by-finite and so in particular chordal graph groups (right-angled Artin groups).