Trivalent vertex-transitive graphs with infinite vertex-stabilizers

TL;DR

This paper classifies connected, vertex-transitive, trivalent graphs with infinite vertex-stabilizers, showing that edge-transitive cases are trees and fully characterizing certain 2-ended graphs, linking to Willis' scale function.

Contribution

It provides a complete classification of such graphs, especially in non-edge-transitive cases, and connects these findings to existing concepts like Willis' scale function.

Findings

Edge-transitive graphs are trees.

Complete classification of 2-ended graphs.

Connections to Willis' scale function and Trofimov's result.

Abstract

We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is not edge-transitive and fully classify the possible $2$-ended graphs. We draw connections to Willis' scale function and re-prove a result by Trofimov.