Three local actions in 6-valent arc-transitive graphs
Hujdurović, Ademir, 1987- ; Potočnik, Primož, 1971- ; Verret, Gabriel
It is known that there are precisely three transitive permutation groups of degree ▫$6$▫ that admit an invariant partition with three parts of size ▫$2$▫ such that the kernel of the action on the ...parts has order ▫$4$▫; these groups are called ▫$A_4(6)$▫, ▫$S_4(6d)$▫ and ▫$S_4(6c)$▫. For each ▫$L\in \{A_4(6), S_4(6d), S_4(6c)\}$▫, we construct an infinite family of finite connected ▫$6$▫-valent graphs ▫$\{\Gamma_n\}_{n\in \mathbb{N}}$▫ and arc-transitive groups ▫$G_n \le \rm{Aut}(\Gamma_n)$▫ such that the permutation group induced by the action of the vertex-stabiliser ▫$(G_n)_v$▫ on the neighbourhood of a vertex ▫$v$▫ is permutation isomorphic to ▫$L$▫, and such that ▫$|(G_n)_v|$▫ is exponential in ▫$|\rm{V}(\Gamma_n)|$▫. These three groups were the only transitive permutation groups of degree at most ▫$7$▫ for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic ▫$2$▫-arc-transitive graphs such that the dimension of the ▫$1$▫-eigenspace over the field of order ▫$2$▫ of the adjacency matrix of the graph grows linearly with the order of the graph.
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