The infinitude of locally 9-arc-transitive graphs

A graph Γ is called locally s-arc-transitive if the stabiliser in Aut ( Γ ) of a vertex v is transitive on the set of all r -arcs in Γ with initial vertex v , for every r ≤ s . A theorem by Stellmacher and van Bon (2015) states that if Γ is a connected finite locally s -arc-transitive graph in which every vertex has valency at least 3, then s ≤ 9 . This theorem complements Tutte's famous theorem for s -arc-transitive finite graphs of valency 3 (showing that s ≤ 5 ) and its extension by Weiss to s -arc-transitive finite graphs of higher valency (for which s ≤ 7 ). In the current paper, the author gives a positive answer to a question by Michael Giudici, by showing that locally 9-arc-transitive graphs are not as rare as might have been expected. Specifically, it is proved that for all but finitely many n , there exists a finite graph upon which the alternating group A n acts as a locally 9-arc-transitive group of automorphisms. 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.