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

Abstract

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

Bounding the covolume of lattices in products

“…As mentioned in § 6.1, it is conjectured that this is always the case when the action is locally semiprimitive (so, conjecturally, assumption (3) automatically implies assumption (4)). This has been confirmed for numerous specific classes of local actions, including -transitive groups [TW95], primitive groups of affine type [Wei79, Spi16], primitive groups of twisted wreath type [Spi11] or semiprimitive groups with two distinct minimal normal subgroups [GM18].…”

Arc-transitive Cayley graphs on nonabelian simple groups with prime valency

“…On locally primitive graphs, Weiss [23] conjectured that there is a function f defined on the positive integers such that, whenever Γ is a G-vertex-transitive locally primitive graph of valency d with G ≤ Aut(Γ ) then, for any vertex v ∈ V (Γ ), |G v | ≤ f (d). By Conder et al [1], Weiss conjecture is true for vertex-transitive locally primitive d-valent graphs if d ≤ 20 or d is a prime number, and by Spiga [21], Weiss conjecture is also true if the restriction G Γ (v) of G on Γ (v) contains an abelian regular subgroup, that is, of affine type. In 2007, Fang et al [8,Theorem 1.1] shown that for any valency d for which the Weiss conjecture holds, all but finitely many locally primitive Cayley graphs of valency d on the finite nonabelian simple groups are normal, and based on this, the following problem was proposed: Clearly, a tetravalent graph is locally primitive if and only if the graph is 2-arctransitive.…”

“…In particular, Aut(Γ ) v is solvable for v ∈ V (Γ ). Let x, y, t, z be permutations in S 30 as following: x = (1,21,10,9,22,28,13,15,30,6,19,18,7,27,23,4,25,17,20,2,12,29,16,26,8,11,3,24,5) y = (1, 24, 9)(2, 6, 5) (3,27,21) (4,12,20) (7,25,26) (8,10,13) (11,14,16) (15,30,23) (3,7,19,20,4,24,...…”

“…Since α ∈ Aut(T, H, D), we have H b = H and D b = D, and since α ∈ H, we have b ∈ H. It follows that H b is a subgroup of T containing H, and by Atlas [3], H b ∼ = Z 23 : Z 11 . Since H ≤ Aut(T, H, D), we may choose b such that b has order 11, and by Magma, we may let b =(2,14,18,7,16,6,9,20,8,3,4)(5,21,13,22,12,15,11,19,17,23,10) because H = x with x =(1,4,6,7,2,19,3,11,9,20,13,23,16,8,21,5,14,22,18,15,17,10,12). However, D b = (HtH) b = HtH by Magma, a contradiction.…”

Three local actions in 6‐valent arc‐transitive graphs

“…. Tutte's result [28] can then be rephrased as saying that transitive groups of degree 3 are graph-restrictive (with corresponding constant 16), while the famous Weiss conjecture [31] claims that every primitive permutation group is graph-restrictive; see [17] for a survey of results on graph-restrictiveness and [2,9,10,26] for more recent results.…”