On Benjamini-Schramm limits of congruence subgroups
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ON BENJAMINI–SCHRAMM LIMITS OF CONGRUENCE SUBGROUPS BY
Arie Levit Department of Mathematics, Yale University, New Haven, CT 06511, USA e-mail: [email protected]
ABSTRACT
A sequence of orbifolds corresponding to pairwise non-conjugate congruence lattices in a higher rank semisimple group over zero characteristic local fields is Benjamini–Schramm convergent to the universal cover.
1. Introduction A semisimple analytic group G is defined as follows. Let I be a finite index set. Assume that ki is a zero characteristic local field and Gi is a connected simply-connected ki -isotropic almost ki -simple linear ki -group for every i ∈ I. Denote Gi = Gi (ki ) so that in particular Gi is an almost simple non-compact linear group admitting a ki -analytic structure. Let G = i∈I Gi . Definition: A sequence of lattices (Γn )n∈N in G is called weakly central1 if for every compact subset Q ⊂ G we have that n→∞
ηn ({gΓn ∈ G/Γn : gΓn g −1 ∩ Q ⊂ Z(G)}) −−−−→ 1 where ηn is the G-invariant probability measure on G/Γn for each n ∈ N. This note is dedicated to establishing the following result. Theorem 1: Assume that |I| ≥ 2. Then every sequence of pairwise nonconjugate congruence lattices in G is weakly central. Received June 11, 2018 and in revised form July 27, 2019 1 Of course, this definition makes sense for any locally compact group.
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A. LEVIT
Isr. J. Math.
Recall that by a celebrated theorem of Margulis every irreducible lattice in G is arithmetic whenever |I| ≥ 2. A congruence lattice is a particular kind of an irreducible arithmetic lattice containing a principal congruence subgroup. See §3 for a precise definition of this notion. In particular, whenever lattices in G are known to satisfy the congruence subgroup property a stronger formulation of Theorem 1 is made possible. We remark that if |I| = 1 and rank(G) ≥ 2 then every sequence of pairwise non-conjugate lattices is weakly central by the main results of [1, 12]. The recent works of Raimbault [21] and Fraczyk [10] establish closely related results for congruence lattices in the rank one groups SL2 (R) and SL2 (C). We also mention [1, §5] dealing with congruence subgroups in a fixed uniform arithmetic lattice. Benjamini–Schramm convergence. The semisimple analytic group G is acting by isometries on a contractible non-positively curved metric space X, as follows. Let Xi be the symmetric space or Bruhat–Tits building associated to Gi for every i ∈ I, depending on whether ki is Archimedean or not. Take X = i∈I Xi equipped with the product metric. Let (Γn ) be a sequence of lattices in G. The following geometric notion is equivalent to saying that (Γn ) is weakly central. See [12, §3] for more details. Definition: The orbifolds Γn \X Benjamini–Schramm converge to X if for every radius 0 < R < ∞ the probability that an R-ball in Γn \X with base point taken uniformly at random is contractible tends to one as n → ∞. As an example, we provide a geometric application of Theorem 1 to arithmetic orbifolds, relying on the congruence subgroup property [23]. Corollary 2: Let F b
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