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

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