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

Abstract

Suppose that \(\Delta \) is 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 \(G_z^{[1]}\) the kernel of the action of \(G_z\) on the neighbours of z. We say \(\Delta \) is of pushing up type provided there exist 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 \( \textrm{PSL}_2(p^a)\le X \le \mathrm{P\Gamma L}_2(p^a)\).