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Schreier graphs: transitivity and coverings

Paul-Henry Leemann

TL;DR

The paper establishes a rigidity-type framework for Schreier graphs by linking isomorphisms to the underlying group/subgroup/generating-system data, yielding a Mostow-like result for $2d$-regular graphs. It derives a precise transitivity criterion: a Schreier graph is vertex-transitive exactly when its subgroup is length-transitive, and it proves a full characterization of when Schreier graphs are isomorphic (root-preserving) in terms of length-preserving subgroup correspondences. The work introduces length-transitive and strongly simple groups, with Tarski monsters as key infinite examples, and uses these concepts to analyze coverings and quasi-isometries, addressing open questions of graph coverings in the Benjamini framework. Together, these results deepen the connection between Schreier graphs and group-theoretic structure, informing both rigidity phenomena and covering theory in geometric group theory.

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.

Schreier graphs: transitivity and coverings

TL;DR

The paper establishes a rigidity-type framework for Schreier graphs by linking isomorphisms to the underlying group/subgroup/generating-system data, yielding a Mostow-like result for -regular graphs. It derives a precise transitivity criterion: a Schreier graph is vertex-transitive exactly when its subgroup is length-transitive, and it proves a full characterization of when Schreier graphs are isomorphic (root-preserving) in terms of length-preserving subgroup correspondences. The work introduces length-transitive and strongly simple groups, with Tarski monsters as key infinite examples, and uses these concepts to analyze coverings and quasi-isometries, addressing open questions of graph coverings in the Benjamini framework. Together, these results deepen the connection between Schreier graphs and group-theoretic structure, informing both rigidity phenomena and covering theory in geometric group theory.

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.

Paper Structure

This paper contains 6 sections, 30 theorems, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. Basic facts about Schreier graphs
  4. A criterion for transitivity
  5. Coverings
  6. Application to groups

Key Result

Lemma 3.1

Let ${\mathcal{H}}$ be any subgroup of ${\mathcal{G}}$ and $\Gamma\coloneqq\mathop{\mathrm{Sch}}\nolimits(\mathcal{G},\mathcal{H},X^\pm)$ be the corresponding Schreier graph. Then, for every vertex $v$ in $\Gamma$, there is a bijection between reduced paths starting at $v$ and elements of ${\mathcal

Figures (6)

  • Figure 1: The Petersen graph viewed as a Schreier graph on $\langle x,a\mid a^2\rangle\simeq\mathbf{Z}*\mathbf{Z}/2\mathbf{Z}$.
  • Figure 2: A non-transitive Schreier graph over $\langle a,x\mid a^2\rangle$. The root is marked in black.
  • Figure 3: An $X$-covering between two Schreier graphs over the free group of rank two. The root of the base graph and its two preimages are marked in black.
  • Figure 4: An $X$-covering between two Schreier graphs over $\langle x,a\mid a^2\rangle$. The root of the base graph and its two preimages are marked in black.
  • Figure 5: An $X$-covering between two Schreier graphs over the free group of rank two. The root of the base graph and its two preimages are marked in black.
  • ...and 1 more figures

Theorems & Definitions (70)

  • Lemma 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Remark 4.1
  • Theorem 4.1
  • Theorem 4.2: Rigidity theorem for regular graphs
  • Proposition 4.1
  • proof
  • ...and 60 more