Context-free graphs and their transition groups

TL;DR

This paper introduces a new class of groups derived from context-free inverse graphs, exploring their properties, closure, and structural constraints, with implications for group theory conjectures and graph modifications.

Contribution

It defines and analyzes a novel class of groups from context-free graphs, establishing their properties and connections to existing group classes and conjectures.

Findings

Transition groups have context-free co-word problems.

Closure under generalized free products preserves context-freeness.

Local graph modifications influence global group structure.

Abstract

Starting from context-free inverse graphs, we introduce a new class of groups and study their structural properties. We establish closure properties, show that their co-word problems are context-free, analyze torsion elements, and realize them as subgroups of the asynchronous rational group. Context-freeness is preserved under a generalized free product of graphs, and using this construction we provide examples of groups that are not residually finite or not poly-context-free, making them relevant for testing the Lehnert and Brough conjectures. Moreover, we investigate how small local modifications of a graph affect the global structure of the transition group, showing that for locally quasi-transitive graphs with infinite orbits, the transition group decomposes into a highly structured quotient by a bounded torsion subgroup, showing strong global constraints induced by local graph properties.