An application of the Local C(G,T) Theorem to a conjecture of Weiss

TL;DR

This paper proves Weiss's conjecture for certain vertex-transitive graphs by applying the Local C(G,T) Theorem, showing the stabilizer of a radius-4 ball is trivial, thus bounding the stabilizer size.

Contribution

It applies the Local C(G,T) Theorem to confirm Weiss's conjecture when the local action contains an abelian regular subgroup.

Findings

Proves Weiss's conjecture under specific conditions.

Shows the point-wise stabilizer of a radius-4 ball is trivial.

Provides bounds on vertex stabilizer sizes.

Abstract

Let $\Gamma$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $\Gamma$ and let $G_v^{\Gamma(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. The graph $\Gamma$ is said to be $G$-\emph{locally primitive} if $G_v^{\Gamma(v)}$ is primitive. Richard Weiss conjectured in $1978$ that, there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that, if $\Gamma$ is a connected $G$-vertex-transitive locally primitive graph of valency $d$ and $v$ is a vertex of $\Gamma$ with $|G_v|$ finite, then $|G_v|\leq f(d)$. As an application of the Local $C(G,T)$ Theorem, we prove this conjecture when $G_v^{\Gamma(v)}$ contains an abelian regular subgroup. In fact, we show that the point-wise stabiliser in $G$ of a ball of $\Gamma$ of radius $4$ is the identity subgroup.