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On finite approximations of transitive graphs

Andreas Thom

Abstract

In this note we answer a question of Johannes Carmesin, which was circulated at the Oberwolfach Workshop on "Graph Theory" in January 2025. We provide a unimodular, locally finite, and vertex-transitive graph without any perfect finite $r$-local model for $r \in \mathbb N$ large enough.

On finite approximations of transitive graphs

Abstract

In this note we answer a question of Johannes Carmesin, which was circulated at the Oberwolfach Workshop on "Graph Theory" in January 2025. We provide a unimodular, locally finite, and vertex-transitive graph without any perfect finite -local model for large enough.

Paper Structure

This paper contains 1 section, 3 theorems, 1 equation.

Table of Contents

  1. Acknowledgments

Key Result

Lemma 3

Let $\Gamma$ be a finitely generated group with a finite generating set $S=S^{-1}$ with $1 \not \in S$, such that ${\rm Cay}(\Gamma,S)$ is automorphism-regular. For every $r \in \mathbb N$, there exists $r_0:=r_0(r) \in \mathbb N$, such that any rooted automorphism of $B_G(e,r_0)$ fixes $B_G(e,r)$ p

Theorems & Definitions (9)

  • Remark 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Corollary 6
  • proof
  • Remark 7