An application of the Local C(G, T) Theorem to a conjecture of Weiss

作者：Spiga Pablo^*

来源：Bulletin of the London Mathematical Society, 2016, 48(1): 12-18.

DOI：10.1112/blms/bdv071

Let Gamma be a connected G-vertex-transitive graph, let v be a vertex of Gamma and let G(v)(Gamma(v)) be the permutation group induced by the action of the vertex-stabilizer G(v) on the neighbourhood Gamma(v). The graph Gamma is said to be G-locally primitive if G(v)(Gamma(v)) is primitive.
Weiss conjectured in 1978 that there exists a function f : N -> N such that, if Gamma is a connected G-vertex-transitive locally primitive graph of valency d and v is a vertex of Gamma with vertical bar Gv vertical bar finite, then vertical bar G(v)vertical bar <= f(d). As an application of the Local C(G, T) Theorem, we prove this conjecture when G(v)(Gamma(v)) contains an abelian regular subgroup. In fact, we show that the point-wise stabilizer in G of a ball of G of radius 4 is the identity subgroup.

• 出版日期2016-2