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

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.