The Weiss conjecture for primitive local actions of almost simple, simple diagonal, and product action type

The Weiss conjecture, posed by Richard Weiss in 1978, predicts that the vertex stabilizer of a connected, vertex-transitive, locally primitive graph is bounded by a function of the valency alone. It is settled when the primitive local action is 2-transitive (Weiss and Trofimov), of affine type (Weiss 1979 and Spiga 2016, via the Local C(G,T) Theorem), and of twisted wreath type (Spiga 2011). Spiga's 2016 completion of the affine case records that the almost simple, simple diagonal, and product action types of the O'Nan-Scott classification remain open. The surviving core is a precise prove-or-refute question: does a valency-only bound on vertex stabilizers hold for locally primitive graphs whose primitive local action is of almost simple (non-2-transitive), simple diagonal, or product action type?

Background

Let $\Gamma$ be a connected, locally finite graph and let $G\leq \mathrm{Aut}(\Gamma)$ act transitively on the vertices; one then says that $\Gamma$ is $G$-vertex-transitive. The stabilizer $G_v$ of a vertex $v$ permutes the set $\Gamma(v)$ of neighbors of $v$, and the permutation group $G_v^{\Gamma(v)}$ induced by this action is called the local action of $G$ at $v$. The pair $(\Gamma,G)$ is locally primitive when $G_v^{\Gamma(v)}$ is a primitive permutation group, that is, it preserves no nontrivial partition of $\Gamma(v)$. Since a primitive group is in particular transitive on $\Gamma(v)$, a locally primitive graph is automatically arc-transitive (transitive on ordered adjacent pairs of vertices). The guiding question of this area, going back to Tutte's 1947 theorem that a finite connected cubic arc-transitive graph has $|G_v|\leq 48$, is how large the vertex stabilizer $G_v$ can be in terms of the valency $d=|\Gamma(v)|$ alone. A closely related classical question is the Sims conjecture for a finite primitive permutation group with a suborbit (an orbit of a point stabilizer) of length $d>1$: Sims conjectured that the point stabilizer then has order bounded by a function of $d$; this was later proved using the classification of finite simple groups (CFSG). In the Szeged 1978 conference proceedings, Richard Weiss posed the local analogue for graphs: there exists $f:\mathbb{N}\to\mathbb{N}$ such that every connected $G$-vertex-transitive locally primitive graph of valency $d$ with finite vertex stabilizers satisfies $|G_v|\leq f(d)$. Equivalently, for such graphs $|\mathrm{Aut}(\Gamma)|\leq f(d)\,|V\Gamma|$; the stabilizer formulation is the right one, since $|\mathrm{Aut}(\Gamma)|$ alone cannot be bounded by the valency (a cycle of valency 2 is locally primitive with $|\mathrm{Aut}|=2|V\Gamma|$).

A finite primitive permutation group of degree $d$ belongs, by the O'Nan-Scott classification as formulated by Liebeck-Praeger-Saxl, to one of five types determined by the socle (the product of the minimal normal subgroups): HA (affine: an elementary abelian $p$-group is a regular normal subgroup, so $d$ is a prime power), AS (almost simple: the socle is a nonabelian simple group $T$ with $T\leq G\leq \mathrm{Aut}(T)$), SD (simple diagonal: the socle is $T^{\ell}$ acting on the cosets of a diagonal subgroup, of degree $|T|^{\ell-1}$), PA (product action: a wreath product with almost simple base acting coordinatewise on tuples), and TW (twisted wreath: the socle $T^{\ell}$ is a nonabelian regular normal subgroup).

The Weiss conjecture is known for local actions of the following kinds. By a long series of papers of Weiss (1979-1993) and Trofimov (1990-2003), the conjecture holds when the local action is 2-transitive (transitive on ordered pairs of distinct neighbors), which covers many almost simple local actions. Spiga (2011) settled the TW type, showing that the subgroup $G_{uv}^{[1]}$ fixing an arc $(u,v)$ and both neighborhoods pointwise is trivial. For the HA type, Weiss (1979) showed that either $G_{uv}^{[1]}=1$ or the socle of the local action is an elementary abelian 2- or 3-group, and Spiga (2016) eliminated the remaining cases as an application of the Local $C(G,T)$ Theorem of Bundy-Hebbinghaus-Stellmacher, a deep result of local finite group theory controlling subgroups of a finite group from the structure of a proper subgroup: if the local action is primitive of HA type and $d=r^{\ell}$, then the pointwise stabilizer of a ball of radius 4 is trivial. What remains is exactly the local actions of almost simple type that are not 2-transitive, together with the simple diagonal and product action types, which are never 2-transitive. A useful reformulation is due to Potočnik-Spiga-Verret: a transitive permutation group $L$ is graph-restrictive if $|G_v|$ is bounded by a constant $c(L)$ over all locally-$L$ pairs $(\Gamma,G)$ (pairs whose local action is permutation isomorphic to $L$); the Weiss conjecture is equivalent to the assertion that every finite primitive permutation group is graph-restrictive (for fixed $d$ there are only finitely many primitive groups of degree $d$ up to conjugacy, so per-group constants combine into a function of $d$). Potočnik-Spiga-Verret proved that every graph-restrictive group is semiprimitive (every normal subgroup is transitive or semiregular, i.e. has trivial point stabilizers) and conjectured the converse (the PSV conjecture), which would imply both the Weiss conjecture and its quasiprimitive analogue due to Praeger. The PSV conjecture is proved only for special classes of semiprimitive groups; primitive groups of AS, SD, and PA type have a unique minimal normal subgroup and so fall outside the verified classes. Reduction theorems complement the case analysis: Conder-Li-Praeger (2000) reduced the non-bipartite case to graphs whose automorphism group contains a locally primitive almost simple subgroup; Praeger-Spiga-Verret (2012) reduced, via normal quotients, to quasiprimitive or biquasiprimitive acting groups and ultimately to nonabelian simple groups; Praeger-Pyber-Spiga-Szabó (2012) confirmed the conjecture when all composition factors of the group have bounded rank.

Problem Statement

Resolve the Weiss conjecture for the primitive local-action types left open by the settled cases. Precisely: does there exist a function $f:\mathbb{N}\to\mathbb{N}$ such that, for every connected, locally finite graph $\Gamma$ of valency $d$ admitting a vertex-transitive group $G\leq\mathrm{Aut}(\Gamma)$ with finite vertex stabilizers whose local action $G_v^{\Gamma(v)}$ is a primitive permutation group of almost simple type (where the 2-transitive subcase is already settled, so the new content is the non-2-transitive almost simple local actions), of simple diagonal type, or of product action type in the O'Nan-Scott classification, the vertex stabilizer satisfies $|G_v|\leq f(d)$? Equivalently: is every finite primitive permutation group of almost simple, simple diagonal, or product action type graph-restrictive, in the sense that $|G_v|$ is bounded, over all locally-$L$ pairs $(\Gamma,G)$, by a constant depending only on $L$? An affirmative answer must exhibit the bound and prove it uniformly in the valency alone (no dependence on $|V\Gamma|$, $|G|$, or the isomorphism type of the local action beyond its degree); a negative answer must produce, for one fixed valency $d$, an infinite family of connected, vertex-transitive, locally primitive graphs with primitive local action of one of these types whose vertex stabilizers have unbounded order - a single finite instance does not refute the conjecture. If Trofimov's 2022 announcement for the simple diagonal type is confirmed, the remaining content is the almost simple and product action types; the almost simple case is the expected crux, since the non-bipartite case reduces to it (Conder-Li-Praeger 2000).

References

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