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Vertex-primitive \(s\)-arc-transitive digraphs of almost simple groups

陈磊 Lei Chen

University of Western Australia

Time: 16:00-17:00 (GMT+8), Wednesday March 13th, 2024
Location: Zoom

Abstract: The property of \(s\)-arc-transitivity has been studied for decades. In 1947 Tutte showed that a cubic graph can be at most \(5\)- arc-transitive. Weiss proved that finite undirected graphs of valency at least \(3\) that are not cycles can be at most \(7\)-arc-transitive. In stark contrast with the situation in undirected graphs, Praeger showed that for each \(s\) and \(d\) there are infinitely many finite \(s\)-arc-transitive digraphs of valency \(d\) that are not \((s + 1)\)-arc-transitive. However, once we add the constraint of primitivity things can get quite different. The vertex-primitive \(s\)-arc-transitive digraphs for large s seem rare. Although extensive attempts had been made to construct a vertex-primitive \(s\)- arc-transitive digraph for \(s \ge 2\), no such examples were found until Giudici, Li, and Xia, found infinite families of vertex-primitive \(2\)-arc-transitive examples, and Giudici and Xia also ask the following question: Question 1 if there exists an upper bound on s for vertex-primitive \(s\)-arc- transitive digraphs that are not directed cycles. Giudici and Xia managed to show that to determine the value of the upper bound in Question 1 it suffices to look at the case where the vertex- primitive automorphism group is almost simple. My Ph.D. project investigates the upper bound on s for \(Sp(2n,q)\), \(G_2(q)\), \( ^3D_4(q)\), \( ^2G_2(q)\), \( ^2F_4(q)\), \( ^2B_2(q)\), and certain types of \(PSU(n,q)\). It turns out that \(s\) is at most \(2\) for those groups.

Host: 丁兆宸 Zhaochen Ding