Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups

AbstractWe prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the {\mathrm{II}_{1}} factor {L(\Gamma)} is prime. In particular, we deduce that the {\mathrm{II}_{1}} factors associated to the arithmetic groups {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer {d\geq 2} with {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for {L(\Gamma)}.

Download Full-text

Strong property (T) for higher-rank simple Lie groups: Figure 1.

Proceedings of the London Mathematical Society ◽

10.1112/plms/pdv040 ◽

2015 ◽

Vol 111 (4) ◽

pp. 936-966 ◽

Cited By ~ 6

Author(s):

Tim de Laat ◽

Mikael de la Salle

Keyword(s):

Lie Groups ◽

Simple Lie Groups ◽

Strong Property ◽

Property T ◽

Higher Rank

Download Full-text

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

International Mathematics Research Notices ◽

10.1093/imrn/rny080 ◽

2018 ◽

Vol 2020 (9) ◽

pp. 2588-2625

Author(s):

Miklos Abert ◽

Nicolas Bergeron ◽

Ian Biringer ◽

Tsachik Gelander ◽

Nikolay Nikolov ◽

...

Keyword(s):

Asymptotic Behavior ◽

Lie Groups ◽

Symmetric Spaces ◽

Betti Numbers ◽

Real Rank ◽

Spectral Invariants ◽

Locally Symmetric Spaces ◽

Rank One ◽

Higher Rank ◽

Invariant Random Subgroups

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).

Download Full-text

Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385701001109 ◽

2001 ◽

Vol 21 (1) ◽

pp. 121-164 ◽

Cited By ~ 15

Author(s):

GREGORY A. MARGULIS ◽

NANTIAN QIAN

Keyword(s):

Lie Groups ◽

Left Translation ◽

Real Rank ◽

Semisimple Lie Groups ◽

Rigidity Result ◽

Global Rigidity ◽

Local Rigidity ◽

Algebraic Actions ◽

Anosov Actions ◽

Higher Rank

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

Download Full-text