On the Growth of L2-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One

Abstract

In the first paper of this series we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statments about spectral invariants of lattices.When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n, 1), n ≥ 2. We give dimension-specific constructions when n = 2, 3, and also describe a general gluing construction that works for every n. Part of the latter contruction is inspired by Gromov and Piatetski-Shapiro's construction of nonarithmetic lattices in SO(n, 1).

Eigenfunctions and Random Waves in the Benjamini-Schramm limit

“…In rank 1 where one can construct many interesting IRSs, e.g. associated to normal subgroups of a lattice (see [1] for more examples), one can similarly construct measures on D ′ (B) from eigenwaves on the corresponding unimodular random X-manifolds.…”

“…In general λ 0 is not an eigenvalue of the Laplace operator on L 2 (Γ n \X). However the following Weyl law type result holds (see the corollary of Theorem 22): 1 Proposition 2. Suppose that (M n ) BS-converges toward X.…”

A model for random three-manifolds

“…The dual graphs follow the same distribution as the configuration model, and as this model of graphs BS-converges to the tree (this follows from [6]) we get that the random variable n converges in distribution to ƒ (since the Schreier graph of P =ƒ is obtained from the tree by replacing vertices with Q). Now n is the IRS obtained by induction of n from 0 to PGL 2 .C/, and O 1 by induction of ƒ (see [2,Section 11.1] for the definition of induction). As induction is continuous we get that 0 n converges to O 1 .…”

Finiteness of maximal geodesic submanifolds in hyperbolic hybrids

“…Perhaps most famously, Gromov and Piatetski-Shapiro used cut-and-paste of arithmetic hyperbolic manifolds along codimension-1 geodesic submanifolds to build non-arithmetic hyperbolic manifolds in all dimensions [19]. More recently, variants on their construction first introduced in [1] were used by Gelander and Levit to prove that "most" hyperbolic manifolds in dimension at least 4 are non-arithmetic [17].…”

“…Also note that the manifolds constructed by Gromov and Piatetski-Shapiro are built from two dissimilar buildings blocks (see Section 2.2), so Theorem 1.4 applies. The theorem also applies to the manifolds used to study invariant random subgroups in [1] and to those used by Raimbault [33] and Gelander and Levit [17] in studying growth of the number of maximal lattices in SO.n; 1/. These lattices are all built from subarithmetic pieces in the language of Gromov and Piatetski-Shapiro [19,Question 0.4].…”