Smooth Classification of Cartan Actions of Higher Rank Semisimple Lie Groups and Their Lattices

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)}.

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Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500790 ◽

2020 ◽

pp. 2050079

Author(s):

Zaili Yan ◽

Shaoqiang Deng

Keyword(s):

Lie Groups ◽

Einstein Metrics ◽

Complete Classification ◽

Semisimple Lie Groups ◽

Killing Form ◽

Interesting Phenomenon ◽

Product Metrics ◽

Coset Spaces ◽

Naturally Reductive

A quadruple of Lie groups [Formula: see text], where [Formula: see text] is a compact semisimple Lie group, [Formula: see text] are closed subgroups of [Formula: see text], and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces [Formula: see text], [Formula: see text] and [Formula: see text] are all Einstein. In this paper, we first give a complete classification of the Einstein basic quadruples. We then show that, except for very few exceptions, given any quadruple [Formula: see text] in our list, we can produce new non-naturally reductive Einstein metrics on the coset space [Formula: see text], by scaling the Killing form metrics along the complement of [Formula: see text] in [Formula: see text] and along the complement of [Formula: see text] in [Formula: see text]. We also show that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics. This discloses a new interesting phenomenon which has not been described in the literature.

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Classification of irreducible tempered representations of semisimple Lie groups

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.73.7.2178 ◽

1976 ◽

Vol 73 (7) ◽

pp. 2178-2180 ◽

Cited By ~ 12

Author(s):

A. W. Knapp ◽

G. Zuckerman

Keyword(s):

Lie Groups ◽

Semisimple Lie Groups

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A classification of equivariant gerbe connections

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500013 ◽

2019 ◽

Vol 21 (02) ◽

pp. 1850001

Author(s):

Byungdo Park ◽

Corbett Redden

Keyword(s):

Lie Groups ◽

Lie Group ◽

Existence And Uniqueness ◽

Smooth Manifold ◽

Equivariant Cohomology ◽

Compact Lie Group ◽

Semisimple Lie Groups ◽

Isomorphism Classes ◽

Bundle Gerbe

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

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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.

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