Vertex-transitive non-Cayley graphs with arbitrarily large vertex-stabilizer (Q1272909)

The authors construct an infinite sequence \(\Gamma_n\) of finite vertex-transitive graphs of fixed valence such that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of \(\Gamma_n\) is a strictly increasing function of \(n\). The graph \(\Gamma_n\) is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree \(p>2^{2^n+2}\). The construction uses Sierpiński's gasket to produce generating permutations for the vertex-stabilizer (which is a large 2-group).