Strong and uniform boundedness of groups

Journal of Topology and Analysis ◽

10.1142/s1793525321500497 ◽

2021 ◽

pp. 1-33

Author(s):

Jarek Kędra ◽

Assaf Libman ◽

Ben Martin

Keyword(s):

Lie Groups ◽

Hamiltonian Dynamics ◽

Algebraic Groups ◽

Uniform Boundedness ◽

Arithmetic Groups ◽

Semisimple Lie Groups ◽

Linear Algebraic Groups ◽

Invariant Norm

A group [Formula: see text] is called bounded if every conjugation-invariant norm on [Formula: see text] has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

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Locally compact groups with every isometric action bounded or proper

Journal of Topology and Analysis ◽

10.1142/s1793525319500547 ◽

2018 ◽

Vol 12 (02) ◽

pp. 267-292

Author(s):

Romain Tessera ◽

Alain Valette

Keyword(s):

Lie Groups ◽

Local Field ◽

Algebraic Groups ◽

Locally Compact Group ◽

Locally Compact Groups ◽

Compact Groups ◽

Isometric Action ◽

Locally Compact ◽

Linear Algebraic Groups ◽

Isometric Actions

A locally compact group [Formula: see text] has property PL if every isometric [Formula: see text]-action either has bounded orbits or is (metrically) proper. For [Formula: see text], say that [Formula: see text] has property BPp if the same alternative holds for the smaller class of affine isometric actions on [Formula: see text]-spaces. We explore properties PL and BPp and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix provides new examples of groups with property PL, including nonlinear ones.

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

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2017-0039 ◽

2019 ◽

Vol 2019 (757) ◽

pp. 197-246 ◽

Cited By ~ 3

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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