scholarly journals Hyperbolic geometry and pointwise ergodic theorems

2017 ◽  
Vol 39 (10) ◽  
pp. 2689-2716
Author(s):  
LEWIS BOWEN ◽  
AMOS NEVO
Keyword(s):  
Large Class ◽  
Lie Groups ◽  
Spectral Theory ◽  
Hyperbolic Space ◽  
Hyperbolic Geometry ◽  
Ergodic Theorems ◽  
Real Rank ◽  
New Approach ◽  
Simple Lie Groups ◽  
Rank One

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

scholarly journals Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

2019 ◽  
Vol 2019 (757) ◽  
pp. 197-246 ◽  
Author(s):  
Daniel Drimbe ◽  
Daniel Hoff ◽  
Adrian Ioana
Keyword(s):  
Lie Groups ◽  
Hyperbolic Groups ◽  
Arithmetic Groups ◽  
Semisimple Lie Groups ◽  
Infinite Groups ◽  
Prime Factorization ◽  
Finite Center ◽  
Simple Lie Groups ◽  
Rank One ◽  
Higher Rank

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)}.

scholarly journals Whittaker transforms on real-rank one Lie groups

1990 ◽  
Vol 60 (1) ◽  
pp. 99-128
Author(s):  
Roe Goodman
Keyword(s):  
Lie Groups ◽  
Real Rank ◽  
Rank One
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