Generalizations of the Muller-Schupp theorem and tree-like inverse graphs

TL;DR

This paper extends the Muller-Schupp theorem by establishing new equivalences for quasi-transitive inverse graphs, linking geometric, automorphism, and language-theoretic properties, and applies these to groups with specific word problem structures.

Contribution

It introduces new characterizations of groups related to inverse graphs, including a group-theoretic analog of the Chomsky-Sch"utzenberger theorem, extending previous results to virtually finitely generated subgroups of direct products of free groups.

Findings

Quasi-transitive inverse graphs quasi-isometric to trees are equivalent to having a virtually free automorphism group.

Groups with word problems as intersections of languages accepted by tree-like inverse graphs are characterized.

Extension of Muller-Schupp theorem to broader classes of groups with language-theoretic properties.

Abstract

We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph $\Gamma$, being quasi-isometric to a tree, or context-free (finitely many end-cones types), or having the automorphism group $Aut(\Gamma)$ that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky-Sch\"utzenberger that is fundamental in solving a weaker version of a conjecture of T. Brough which also extends Muller and Schupp' result to the class of groups that are virtually finitely generated subgroups of direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.