Groups with ET0L co-word problem

TL;DR

This paper investigates groups with co-word problems that are ET0L languages, establishing closure properties and including various well-known groups such as virtually free, automata, and Higman-Thompson groups.

Contribution

It proves closure results for groups with co-ET0L co-word problems, expanding the class to include several important groups and their constructions.

Findings

Includes all known co-ET0L groups at the time of writing.

Shows closure under finitely generated subgroups, extensions, products, and wreath products.

Contains virtually free, automata, and Higman-Thompson groups.

Abstract

We study groups whose co-word problems are ET0L languages, which we call coET0L groups, using an automaton based model due to van Leeuwen, and recently studied by Bishop and Elder. In particular we prove a number of closure results for the class of groups with co-word problems in a subclass of `special' ET0L languages; that class of groups contains all groups that we know at the time of writing to be co-ET0L, including all groups that were proved by Holt and R\"over to be stack groups, and hence co-indexed. It includes virtually free groups, bounded automata groups, and the Higman-Thompson groups, together with groups constructed from those using finitely generated subgroups, finite extension, free and direct products, and by taking the restricted standard wreath product of a co-\E group by a finitely generated virtually free top group.