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

Abstract

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

to appear on the Bulletin of the London Math. Society