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

Author: 

Abert, M
Bergeron, N
Biringer, I
Gelander, T
Nikolov, N
Raimbault, J
Samet, I

Publication Date: 

11 May 2018

Journal: 

International Mathematics Research Notices

Last Updated: 

2019-04-17T17:22:28.823+01:00

DOI: 

10.1093/imrn/rny080

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).

Symplectic id: 

670884

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Submitted to ORA: 

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Publication Type: 

Journal Article