Uniform Kazhdan–Margulis neighbourhoods for cocompact lattices with dense projections in products of quasi just-non-compact tdlc groups (Caprace–Le Boudec, Question 5.1)

Caprace and Le Boudec (Compositio Mathematica 155 (2019), 2296–2333) proved that in a product $G=G_1\times\cdots\times G_n$ ($n\ge 2$) of non-discrete, compactly generated, quasi just-non-compact totally disconnected locally compact (tdlc) groups, the cocompact lattices whose projections are dense in every factor become uniformly discrete once an a priori bound is fixed on their number of orbits, equivalently on a compact set $K$ with $G=\Gamma K$. Their Question 5.1 asks whether this bound can be made uniform over all such lattices at once: does a single identity neighbourhood $V$ of $G$ meet every cocompact lattice $\Gamma\le G$ with dense projections trivially? A positive answer would, via Serre's covolume formula, give the Kazhdan–Margulis conclusion for products — a covolume lower bound depending only on $G$. No resolution is known in the literature: recent work of Caprace–Vast constructs ascending chains of cocompact lattices in products of trees with covolume tending to zero, but these lattices have non-discrete projections that are not dense (an infinite strictly ascending chain of cocompact lattices with dense projections cannot exist, by Theorem A), so the question — open even for products of topologically simple factors — survives with its dense-projection hypothesis shown to be essential.

Background

A classical theorem of Kazhdan and Margulis asserts that in a connected semisimple Lie group $G$ without compact factors there is an identity neighbourhood $V$ meeting every lattice trivially; one consequence is that the covolumes of lattices (the total invariant volume of $G/\Gamma$) are bounded below by a positive constant depending only on $G$. The analogous statement fails badly outside the Lie world: Bass and Kulkarni constructed strictly ascending chains of cocompact lattices in the automorphism group $\mathrm{Aut}(T_d)$ of the $d$-regular tree for $d\ge 5$, so covolumes of cocompact lattices there accumulate at $0$ and no uniform identity neighbourhood exists.

Caprace and Le Boudec studied the intermediate regime of products. A locally compact group is totally disconnected locally compact (tdlc) when its identity component is trivial; $\mathrm{Aut}(T_d)$ with the compact-open topology is the prototype. A lattice in a locally compact group $G$ is a discrete subgroup $\Gamma$ such that $G/\Gamma$ carries a finite $G$-invariant measure; the lattice is cocompact when $G/\Gamma$ is compact. For tdlc groups, Serre's covolume formula expresses the covolume of a lattice through the index of its intersection with a compact open subgroup, so an identity neighbourhood $V$ with $V\cap\Gamma=\{1\}$ forces the covolume of $\Gamma$ to be bounded below in terms of $V$ alone. A tdlc group is called quasi just-non-compact when it is non-compact and every closed normal subgroup is either discrete or cocompact; every compactly generated topologically simple tdlc group satisfies this, and by a local-to-global observation of Burger and Mozes so does every non-discrete closed subgroup of the automorphism group of a connected locally finite graph whose local action (the permutation group induced on the neighbours of a vertex) is semiprimitive, meaning transitive with every intransitive normal subgroup acting freely. A lattice $\Gamma$ in a product $G=G_1\times\cdots\times G_n$ is called irreducible, or is said to have dense projections, when the projection of $\Gamma$ to each factor $G_i$ is dense; this matches the standard notion of irreducibility for lattices in products of almost simple algebraic groups. Note that the word is not used uniformly across the literature: some works on products of trees, among them the ascending-chain construction of Reference 2 below, use "irreducible" in a strictly weaker sense that allows projections which are non-discrete but not dense; throughout this task "irreducible" means dense projections on every factor.

For such products, Caprace and Le Boudec proved (Theorem A) that a cocompact irreducible lattice is contained in only finitely many discrete subgroups, and (Theorem B, restated as Theorem 5.10) that for every compact subset $K$ of $G$ there is an identity neighbourhood $V_K$ such that $V_K\cap\Gamma=\{1\}$ for every $K$-cocompact discrete subgroup $\Gamma$ with dense projections; being $K$-cocompact is equivalent to acting with at most $r$ orbits on $G/U$ for a compact open subgroup $U$, with $r$ controlled by $K$. Hence cocompact irreducible lattices with a bounded number of orbits are uniformly discrete and their covolumes form a finite set. The neighbourhood $V_K$, however, a priori depends on $K$, i.e. on the orbit bound, and the paper's Question 5.1 asks whether this dependence can be removed. A positive answer would be a genuine Kazhdan–Margulis theorem for products of quasi just-non-compact tdlc groups: by Serre's covolume formula it would yield a positive lower bound, depending only on $G$, for the covolumes of all cocompact lattices with dense projections in $G$. The authors record that they do not know the answer even when all factors are topologically simple, and point to a related conjecture for semisimple groups in Margulis's book (Conjecture (B) in IX.4.21). Separately, the same paper reduces finiteness statements for lattices acting on products of graphs with semiprimitive local action to the graph-restrictive conjectures of Weiss, Praeger and Potočnik–Spiga–Verret — a finite permutation group $L$ being graph-restrictive when there is a constant $c_L$ such that every connected locally finite graph admitting $L$ as the local action of a discrete vertex-transitive group has vertex stabilizers of order at most $c_L$; those conjectures are independent open problems and are not part of the present question.

Problem Statement

Let $n\ge 2$ and let $G_1,\dots,G_n$ be non-discrete, compactly generated, quasi just-non-compact tdlc groups, and let $G=G_1\times\cdots\times G_n$. Does there exist an identity neighbourhood $V$ in $G$ such that $V\cap\Gamma=\{1\}$ for every cocompact lattice $\Gamma\le G$ whose projection to each factor $G_i$ is dense? Equivalently: can the identity neighbourhood $V_K$ of Theorem B of Caprace–Le Boudec be chosen independently of the compact set $K$ — that is, independently of any a priori bound on the number of orbits of the lattice on $G/U$, where $U$ is any compact open subgroup of $G$? A positive answer would, via Serre's covolume formula, yield the Kazhdan–Margulis conclusion for such products: a constant $c(G)>0$ bounding below the covolume of every cocompact lattice with dense projections in $G$. The question is to be answered in the stated generality (all $n\ge 2$ and all products as above); it is open already when all factors are topologically simple, for instance for products of Burger–Mozes universal groups acting on regular trees.

References

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