Vertex stabilizers of locally $s$-arc transitive graphs of pushing up type

Abstract

Suppose that $\Delta$ a thick, locally finite and locally $s$-arc transitive $G$-graph with $s \ge 4$. For a vertex $z$ in $\Delta$, let $G_z$ be the stabilizer of $z$ and $G_z^{[1]}$ be the kernel of the action of $G_z$ on the neighbours of $z$. We say $\Delta$ is of pushing up type provided there exists a prime $p$ and a $1$-arc $(x,y)$ such that $C_{G_z}(O_p(G_z^{[1]})) \le O_p(G_z^{[1]})$ for $z \in \{x,y\}$ and $O_p(G_x^{[1]}) \le O_p(G_y^{[1]})$. We show that if $\Delta$ is of pushing up type, then $O_p(G_x^{[1]})$ is elementary abelian and $G_x/G_x^{[1]}\cong X$ with ${\rm PSL}_2(p^a)\le X \le {\rm P\Gamma L}_2(p^a)$.