TY - JOUR
T1 - On the Growth of L-2-Invariants of Locally Symmetric Spaces, II
T2 - Exotic Invariant Random Subgroups in Rank One
AU - Abert, Miklos
AU - Bergeron, Nicolas
AU - Biringer, Ian
AU - Gelander, Tsachik
AU - Nikolov, Nikolay
AU - Raimbault, Jean
AU - Samet, Iddo
N1 - The authors would like to thank the referee for a careful reading of the paper, and for the helpful comments. The authors would like to thank Yair Minsky for an invaluable conversation that led to Example 4. This research was supported by the Magyar Tudományos Akadémia Renyi “Lendulet” Groups and Graphs Research Group, the National Science Foundation, the Institut Universitaire de France, the European Research Council Consolidator [grant 648017]; the Israel Science Foundation; the Binational Science Foundation; and the Engineering and Physical Sciences Research Council.
PY - 2020/5
Y1 - 2020/5
N2 - 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).
AB - 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).
U2 - https://doi.org/10.1093/imrn/rny080
DO - https://doi.org/10.1093/imrn/rny080
M3 - مقالة
SN - 1073-7928
VL - 2020
SP - 2588
EP - 2625
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 9
ER -