On the Growth of L2-Invariants of Locally Symmetric Spaces, II: Exotic Invariant Random Subgroups in Rank One. (11th May 2018) - Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo
Abstract: In the 1st 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 statements 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 construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO( n, 1).