The Weiss conjecture for locally primitive graphs with almost simple or product action local type

Determine whether the 1978 Weiss conjecture on vertex stabilizers of locally primitive graphs holds when the primitive local action is of almost simple type or of product action type. Spiga completed the affine case via the Local C(G,T) theorem, Praeger-Pyber-Spiga-Szabo handled automorphism groups whose composition factors have bounded rank, and Trofimov confirmed all O'Nan-Scott types except AS and PA using the Meierfrankenfeld-Stellmacher general FF-module theorem. The problem asks for a proof or refutation of the conjecture when the primitive local action is of almost simple type or of product action type; equivalently, whether every finite primitive permutation group of these two types is graph-restrictive. A resolution would complete the Weiss conjecture and the primitive case of the semiprimitivity conjecture of Potocnik-Spiga-Verret.

Background

Let $\Gamma$ be a finite graph and let $G\leq\mathrm{Aut}(\Gamma)$. The graph is $G$-vertex-transitive when $G$ permutes the vertices transitively; its valency $d$ is the common number of neighbours of a vertex. The vertex stabilizer $G_v$ is the subgroup of $G$ fixing a chosen vertex $v$, and it permutes the set $\Gamma(v)$ of neighbours of $v$; the induced permutation group $G_v^{\Gamma(v)}$ is called the local action of $G$ at $v$. The pair $(\Gamma,G)$ is $G$-locally primitive when this local action is a primitive permutation group on $\Gamma(v)$, that is, it preserves no nontrivial partition of $\Gamma(v)$. Locally primitive graphs generalize $s$-arc-transitive graphs (graphs whose group is transitive on directed walks of length $s$): for $s\geq 2$ the local action is 2-transitive, hence primitive. Tutte proved in 1947 that a connected $G$-vertex-transitive graph of valency $3$ with $G$ transitive on arcs has $|G_v|\leq 48$, the first evidence that the local action can control the size of the whole vertex stabilizer. Motivated by his study of $s$-transitive graphs, Richard Weiss conjectured in 1978 the general phenomenon: there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every finite connected $G$-vertex-transitive, $G$-locally primitive graph of valency $d$ satisfies $|G_v|\leq f(d)$. This is the locally-primitive analogue of the Sims conjecture for primitive permutation groups, which predicts that the order of a point stabilizer in a finite primitive permutation group is bounded by a function of the length of a nontrivial orbit of the point stabilizer (a subdegree). Progress on the conjecture is organized by the O'Nan-Scott theorem, which classifies finite primitive permutation groups into structural types determined by their socle (the product of their minimal normal subgroups): HA or affine type (a subgroup of an affine group with a regular elementary abelian normal subgroup), AS or almost simple type (squeezed between a nonabelian finite simple group $T$ and its automorphism group $\mathrm{Aut}(T)$), SD or simple diagonal type and CD or compound diagonal type (diagonal-action types with single or compounded simple factors, the second distinguished only in finer versions of the theorem), PA or product action type (a wreath product acting in its product action on a Cartesian power), and TW or twisted wreath type (having a regular nonabelian normal subgroup). The type of the local action $G_v^{\Gamma(v)}$ dictates which techniques apply. Potocnik-Spiga-Verret recast the conjecture in the language of graph-restrictive groups: a permutation group $L$ is graph-restrictive if there is a constant $c(L)$ bounding $|G_v|$ for every locally-$L$ pair $(\Gamma,G)$, meaning a finite connected $G$-vertex-transitive graph whose local action is permutation-isomorphic to $L$; the Weiss conjecture is then equivalent to the statement that every finite primitive permutation group is graph-restrictive. By 2011 Weiss's own p-factorization methods had settled the case where the local action is 2-transitive, and Spiga had settled twisted wreath type, obtaining the universal bound $|G_v|\leq d!(d-1)!$ in that case. Since then, the affine case was completed by Spiga using the Local $C(G,T)$ theorem (a local-analysis classification tool originating in the proof of the classification of finite simple groups), automorphism groups all of whose composition factors have bounded rank were handled by Praeger-Pyber-Spiga-Szabo, and Trofimov confirmed every O'Nan-Scott type except AS and PA using the Meierfrankenfeld-Stellmacher general FF-module theorem (a deep local-analysis result on modules with quadratic action). What remains open is almost simple type and product action type; moreover Trofimov constructed a locally projective graph with local group $\mathrm{Aut}(Fi_{22})$ (the almost simple automorphism group of the Fischer sporadic simple group $Fi_{22}$) whose radius-2 ball stabilizer is nontrivial, showing that the vanishing-ball-stabilizer mechanism that settled the other types fails for almost simple local actions. Product action primitive groups are built from an almost simple (or diagonal-type) base by wreath-product inflation.

Problem Statement

Resolve the Weiss conjecture in the two O'Nan-Scott types of local action that remain unsettled. (i) Almost simple type: does there exist a function $f_{AS}:\mathbb{N}\to\mathbb{N}$ such that every finite connected $G$-vertex-transitive, $G$-locally primitive graph $\Gamma$ of valency $d$ whose local action $G_v^{\Gamma(v)}$ is a primitive permutation group of almost simple type (that is, $T\leq G_v^{\Gamma(v)}\leq\mathrm{Aut}(T)$ for a nonabelian finite simple group $T$) satisfies $|G_v|\leq f_{AS}(d)$? (ii) Product action type: the same question with local action primitive of product action type. A positive answer to both, combined with the settled affine, twisted wreath, simple diagonal and compound diagonal cases, would establish the 1978 Weiss conjecture in full and prove that every finite primitive permutation group is graph-restrictive, the primitive case of the semiprimitivity conjecture of Potocnik-Spiga-Verret. A negative answer to either case consists of an infinite family of such graphs of some fixed valency whose vertex stabilizers have unbounded order.

References

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