Bounding the covolume of lattices in products

Abstract

We study lattices in a product G = G 1 ×· · ·×G n of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that G i is non-compact and every closed normal subgroup of G i is discrete or cocompact (e.g. G i is topologically simple).We show that the set of discrete subgroups of G containing a fixed cocompact lattice Γ with dense projections is finite. The same result holds if Γ is nonuniform, provided G has Kazhdan's property (T). We show that for any compact subset K ⊂ G, the collection of discrete subgroups Γ ≤ G with G = ΓK and dense projections is uniformly discrete, hence of covolume bounded away from 0. When the ambient group G is compactly presented, we show in addition that the collection of those lattices falls into finitely many Aut(G)-orbits.As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor.We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-noncompact tdlc group G is a Chabauty limit of discrete subgroups, then some compact open subgroup of G is an infinitely generated pro-p group for some prime p. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

Groups Acting on Trees With Prescribed Local Action

“…We generalize the Burger–Mozes theory of locally quasiprimitive automorphism groups of graphs to the semiprimitive case. While this adjustment of Sections 1.1–1.5 in [2] is straightforward and was initiated in [30, Section II.7] and [3, Section 6.2], we provide a full account for the reader’s convenience.…”

“…https://doi.org/10.1017/S1446788722000143 Published online by Cambridge University Press [3] Groups acting on trees 3 This can be viewed as an interchangeability condition on neighbouring local actions; see Section 4.4. There also is a discreteness condition (D) on F ≤ Aut(B d,k ) in terms of certain stabilizers in F under which U k (F) is discrete; see Section 4.2.2.…”

Three local actions in 6‐valent arc‐transitive graphs

“…This problem was first considered in Tutte's celebrated work [28], where it was proved that, in the 3-valent case, G 16 uv ≤   . Bounding the order of the arc-stabiliser plays a crucial role in many problems, such as when constructing complete lists of graphs of a prescribed symmetry type [3,4,6,18,21] (which have numerous applications themselves), proving asymptotic results regarding the number of graphs of a particular type [22,23], obtaining classification results see [5,7,14,32], or various other problems in algebraic graph theory and elsewhere [2,15,25,27].…”

“…. 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.…”