An Automata Theoretic Approach to the Generalized Word Problem in Graphs of Groups

We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which socalled 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).