On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one

TL;DR

This paper explores the complex space of invariant random subgroups in rank one Lie groups, constructing uncountable families and extending known methods to new geometric and spectral contexts.

Contribution

It introduces new constructions of uncountable invariant random subgroups in SO(n,1), expanding understanding of IRSs in rank one Lie groups beyond higher rank classifications.

Findings

Constructed uncountable families of IRSs in SO(n,1)

Developed dimension-specific IRS constructions for n=2,3

Presented a general gluing construction for all n ≥ 2

Abstract

In the first paper of this series (arxiv.org/abs/1210.2961) 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). We give dimension-specific constructions when n=2,3, and also describe a general gluing construction that works for every n at least 2. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO(n,1).