scholarly journals FUNDAMENTAL DOMAINS FOR LATTICES IN RANK ONE SEMISIMPLE LIE GROUPS

1969 ◽  
Vol 62 (2) ◽  
pp. 309-313 ◽  
Author(s):  
H. Garland ◽  
M. S. Raghunathan
Keyword(s):  
Lie Groups ◽  
Semisimple Lie Groups ◽  
Fundamental Domains ◽  
Rank One

Fundamental Domains for Lattices in (R-)rank 1 Semisimple Lie Groups

1970 ◽  
Vol 92 (2) ◽  
pp. 279 ◽  
Author(s):  
H. Garland ◽  
M. S. Raghunathan
Keyword(s):  
Lie Groups ◽  
Semisimple Lie Groups ◽  
Fundamental Domains

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)}.

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