Tornier's finiteness conjecture for discrete groups acting on trees with semiprimitive local action and an involutive inversion

Let $T_d$ be the $d$-regular tree with $d\ge 3$, and let $F\le\operatorname{Sym}(\Omega)$, $|\Omega|=d$, be a semiprimitive permutation group, i.e. a transitive group in which every normal subgroup is either transitive or semiregular. Tornier (2023, Conjecture 5.41; arXiv numbering 4.41) conjectures that there are only finitely many conjugacy classes of discrete subgroups of $\operatorname{Aut}(T_d)$ which locally act like $F$ and contain an involutive inversion. Equivalently, the invariant $\operatorname{dim}_{CD}(F)$ — the maximal minimal local depth $k$ needed to realize any such group as a generalized Burger–Mozes universal group $\operatorname{U}_k(F^{(k)})$ — is finite for every semiprimitive $F$. The conjecture is a weak form of the classical 1978 Weiss conjecture on locally primitive groups acting on trees, obtained by allowing all semiprimitive local actions but counting only groups that contain an involutive inversion; it is implied by the still-open conjecture of Potočnik–Spiga–Verret that every semiprimitive group is graph-restrictive. Known cases: dimension 1 exactly for regular local actions, dimension 2 for a large class of primitive non-regular local actions, and the cubic case $\operatorname{dim}_{CD}(S_3)=3$; for imprimitive semiprimitive groups such as wreath products, whose dimension is at least 3, no upper bound is known and the conjecture is open.

Background

Let $T_d$ ($d\ge 3$) denote the $d$-regular tree, the connected graph with no cycles in which every vertex has exactly $d$ neighbours. Its automorphism group $\operatorname{Aut}(T_d)$, endowed with the topology of pointwise convergence on vertices, is a totally disconnected locally compact (t.d.l.c.) group. A subgroup $H\le\operatorname{Aut}(T_d)$ is discrete precisely when all of its vertex stabilizers $H_x=\{h\in H:hx=x\}$ are finite; $H$ is vertex-transitive when it acts transitively on the vertex set. The local action of $H$ at a vertex $x$ is the permutation group induced by $H_x$ on the $d$ neighbours of $x$; for a transitive permutation group $F\le\operatorname{Sym}(\Omega)$ with $|\Omega|=d$, one says that $H$ locally acts like $F$ when the local action of $H$ at every vertex is isomorphic to $F$ (so $H$ is locally transitive). An inversion is an automorphism that interchanges the two endpoints of some edge, and an involutive inversion is an inversion of order exactly $2$. A permutation group is semiregular when no non-identity element fixes a point, and regular when it is transitive and semiregular; a transitive group is semiprimitive when each of its normal subgroups is either transitive or semiregular. Primitive and quasiprimitive groups are semiprimitive, but so are many imprimitive groups, such as the action of $S_4$ on the $12$ ordered pairs of distinct elements.

The finiteness question originates with Tutte, who proved in 1947 that a vertex stabilizer in an arc-transitive graph of valency $3$ has order dividing $48$; Djoković–Miller (1980) then showed that, up to conjugacy, there are exactly seven discrete, vertex-transitive, locally transitive subgroups of $\operatorname{Aut}(T_3)$. Weiss (1978) conjectured the general statement: for every $d\ge 3$ there are only finitely many conjugacy classes of discrete, vertex-transitive, locally primitive subgroups of $\operatorname{Aut}(T_d)$, where locally primitive means that every local action is a primitive permutation group. Potočnik–Spiga–Verret (2012) rephrased this using graph-restrictive groups: a transitive group $L$ is graph-restrictive if the arc stabilizers $|G_{uv}|$ are bounded by a constant $c(L)$ across all vertex-transitive graphs locally acting like $L$. They proved that every graph-restrictive group is semiprimitive, and conjectured the converse; the primitive case of this converse is exactly the Weiss conjecture. Partial results toward the converse, all far from the general semiprimitive case, are due to Giudici–Morgan (2014, an infinite family of semiprimitive graph-restrictive groups via the amalgam method; 2015, locally semiprimitive graphs with a regular nilpotent normal subgroup; 2018, a structure theory of semiprimitive groups) and Morgan (2022, the symmetric group acting on ordered pairs).

Burger–Mozes (2000) introduced universal groups as the largest closed subgroups of $\operatorname{Aut}(T_d)$ with prescribed local action $F\le\operatorname{Sym}(\Omega)$: after labelling the edges around each vertex by $\Omega$, $\operatorname{U}(F)$ consists of all automorphisms whose local action at every vertex lies in $F$. Tornier (2023) generalized this to $k$-local actions: $B_{d,k}$ is the ball of radius $k$ in $T_d$, $\operatorname{Aut}(B_{d,k})$ its automorphism group, and for $F\le\operatorname{Aut}(B_{d,k})$ the group $\operatorname{U}_k(F)$ consists of all automorphisms of $T_d$ whose induced action on every $k$-ball lies in $F$. Two conditions make the construction behave: $F$ satisfies (C) when $\operatorname{U}_k(F)$ locally acts like $F$ (a compatibility condition on neighbouring local actions), and (D) is a discreteness condition on stabilizers under which $\operatorname{U}_k(F)$ is discrete. Tornier's Corollary 5.38 (arXiv 4.38) shows that every discrete, locally transitive subgroup of $\operatorname{Aut}(T_d)$ containing an involutive inversion is of the form $\operatorname{U}_k(F^{(k)})$ for some $k$ and some $F^{(k)}\le\operatorname{Aut}(B_{d,k})$ satisfying (C) and (D); since every $\operatorname{U}_k(F^{(k)})$ is vertex-transitive, vertex-transitivity is automatic for the groups being counted. He then defines, for transitive $F$, the invariant $\operatorname{dim}_{CD}(F)$ as the maximum over all discrete $H\le\operatorname{Aut}(T_d)$ locally acting like $F$ with an involutive inversion of the minimal such $k$, and observes that the conjecture is equivalent to finiteness of $\operatorname{dim}_{CD}(F)$ for all semiprimitive $F$. The paper determines $\operatorname{dim}_{CD}(F)=1$ exactly when $F$ is regular; $\operatorname{dim}_{CD}(F)=2$ for primitive non-regular $F$ whose point stabilizers have trivial nilpotent radical (a class covering many almost simple primitive groups); $\operatorname{dim}_{CD}(S_3)=3$, recovering Djoković–Miller; and $\operatorname{dim}_{CD}(F\wr P)\ge 3$ for all non-trivial transitive $F$ and $P$. Thus the reduction turns the finiteness conjecture into a uniformly bounded enumeration of subgroups of the finite groups $\operatorname{Aut}(B_{d,k})$ satisfying (C) and (D), but no such uniform bound is known, and the question is open exactly for semiprimitive groups beyond the dimension-$1$ and dimension-$2$ classes — above all imprimitive semiprimitive groups.

Problem Statement

Let $d\ge 3$ and let $F\le\operatorname{Sym}(\Omega)$, $|\Omega|=d$, be a semiprimitive permutation group, that is, $F$ is transitive and every normal subgroup of $F$ is either transitive or semiregular. Must there be only finitely many $\operatorname{Aut}(T_d)$-conjugacy classes of discrete subgroups $H\le\operatorname{Aut}(T_d)$ such that (i) the local action of $H$ at every vertex is isomorphic to $F$ and (ii) $H$ contains an involutive inversion (an element of order $2$ swapping two adjacent vertices)? Equivalently, is $\operatorname{dim}_{CD}(F)$ finite for every semiprimitive $F$, where $\operatorname{dim}_{CD}(F)$ is the maximum over all such $H$ of the minimal $k$ for which $H=\operatorname{U}_k(F^{(k)})$ for some $F^{(k)}\le\operatorname{Aut}(B_{d,k})$ satisfying conditions (C) and (D)? A positive answer must cover all semiprimitive $F$ of every degree $d\ge 3$, primitive and imprimitive, regular and non-regular; a negative answer consists of a single semiprimitive $F$ together with infinitely many pairwise $\operatorname{Aut}(T_d)$-non-conjugate discrete subgroups locally acting like $F$, each containing an involutive inversion. Vertex-transitivity of $H$ is not an additional hypothesis: it follows from (i) and (ii) via Tornier's Corollary 5.38. This is a derived, weakened form of the 1978 Weiss conjecture (which asserts the same finiteness for locally primitive groups without the inversion hypothesis) and is implied by the open Potočnik–Spiga–Verret conjecture that every semiprimitive group is graph-restrictive; it must not be confused with either stronger statement.

References

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