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

TL;DR

The paper generalizes the Muller–Schupp framework from groups to inverse graphs, establishing that for infinite quasi-transitive inverse graphs, context-free structure, tree-likeness, and virtually free automorphism groups are equivalent, and that a finite-quotient representation via a cover and a surjection from a fundamental group onto a free group underpins this equivalence. It then provides a representation-theoretic analogue to Chomsky–Schützenberger, linking context-free inverse graphs to inverse pushdown automata and showing that the word problem in certain poly-context-free settings corresponds to intersections of languages accepted by tree-like inverse graphs. A key consequence is a characterization of groups virtually embeddable in direct products of free groups as those whose word problem lies in the class PT-ICF, i.e., intersections of languages accepted by quasi-transitive, tree-like inverse graphs. The results merge graph theory, geometric group theory, and automata theory to extend classical Muller–Schupp phenomena to a broad inverse-graph setting with clear group-theoretic implications.

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 $Γ$, being quasi-isometric to a tree, or context-free (finitely many end-cones types), or having the automorphism group $Aut(Γ)$ 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-Schützenberger 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.

Figures (2)

• Figure 1: The first step and the generic $i$-th step of the triangulation of a circuit $p=x_0{\stackrel{\text{\tiny{$w$}}}{\relbar\joinrel\longrightarrow}} x_0$.
• Figure 2: In red the external walks $\mathcal{E}(p, C)$ of the walk $p=v_1{\stackrel{\text{\tiny{$w$}}}{\relbar\joinrel\longrightarrow}}v_2$.

Theorems & Definitions (35)

• Theorem
• Theorem
• Theorem
• Definition 1: involutive and inverse graphs
• Lemma 1
• Lemma 2
• proof
• Proposition 1
• proof
• Proposition 2