Schreier graphs: transitivity and coverings

TL;DR

This paper characterizes isomorphisms between Schreier graphs using group-theoretic data, provides a transitivity criterion, and demonstrates that Tarski monsters meet a strong simplicity condition.

Contribution

It offers a new characterization of Schreier graph isomorphisms, analogous to Mostow's rigidity, and applies it to transitivity and Tarski monsters.

Findings

Characterization of Schreier graph isomorphisms via groups and generating systems

A transitivity criterion for Schreier graphs

Tarski monsters satisfy a strong simplicity criterion

Abstract

We give a characterization of isomorphisms between Schreier graphs in terms of the groups, subgroups and generating systems. This characterization may be thought as a graph analog of Mostow's rigidity theorem for hyperbolic manifolds. This allows us to give a transitivity criterion for Schreier graphs. Finally, we show that Tarski monsters satisfy a strong simplicity criterion.