Marston Conder: The infinitude of locally 9-arc-transitive graphs

The infinitude of locally $s$-arc-transitive graphs

Marston Conder (University of Auckland, New Zealand).

Abstract: A graph $\Gamma$ is said to be locally $s$-arc-transitive if the stabiliser in $\operatorname{Aut}(\Gamma)$ of a vertex $v$ is transitive on the set of all $r$-arcs in $\Gamma$ with initial vertex $v$, for every $r \le s$. It was conjectured by Stellmacher well over 15 years ago that if $\Gamma$ is a connected finite locally $s$-arc-transitive graph in which every vertex has valency at least $3$, then $s \le 9$, and this was proved by Stellmacher and van Bon in 2014/15.

Their theorem complements Tutte's famous theorem for $s$-arc-transitive graphs of valency $3$, and its extension by (Richard) Weiss to $s$-arc-transitive graphs of higher valency.

The first (and smallest) known example meeting the Stellmacher bound $s \le 9$ is a bipartite graph of order $4680$ (with vertices of valency $3$ in one part and $5$ in the other), with automorphism group the exceptional simple simple group Ree ${}^{2}F_4(2)$, constructible from a generalised octagon.

At the PhD summer school at Rogla in 2011, Michael Giudici asked whether other examples could be constructed, apart from this graph and its covers.

In this talk I'll provide a positive answer, by showing that for all but finitely many $n$, there exists such a bipartite graph (again with vertices of valency $3$ in one part and $5$ in the other) with automorphism group isomorphic to the alternating group $A_n$.

The proof involves the construction and combination of finite quotients of an amalgamated product $A *_C B$ where $A$ and $B$ are vertex-stabilisers of orders $12288$ and $20480$ intersecting in an edge-stabiliser of order $4096$.