scholarly journals Relative Property (T) for Topological Groups

2014 ◽  
Vol 05 (19) ◽  
pp. 2988-2993
Author(s):  
Jicheng Tao ◽  
Wen Yan
Keyword(s):  
Topological Groups ◽  
Relative Property ◽  
Property T

scholarly journals On relative property (T) and Haagerup’s property

2011 ◽  
Vol 363 (12) ◽  
pp. 6407-6420 ◽  
Author(s):  
Ionut Chifan ◽  
Adrian Ioana
Keyword(s):  
Relative Property ◽  
Property T

scholarly journals COARSE EMBEDDINGS INTO A HILBERT SPACE, HAAGERUP PROPERTY AND POINCARÉ INEQUALITIES

2009 ◽  
Vol 01 (01) ◽  
pp. 87-100 ◽  
Author(s):  
ROMAIN TESSERA
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Metric Space ◽  
Unitary Representations ◽  
Poincaré Inequalities ◽  
Haagerup Property ◽  
Relative Property ◽  
Poincare Inequalities ◽  
Isometric Actions ◽  
Property T

We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincaré inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a group does not have the Haagerup Property if and only if it has relative property T with respect to a family of probabilities whose supports go to infinity. We give versions of this result both in terms of unitary representations, and in terms of affine isometric actions on Hilbert spaces.

Borel cocycles, approximation properties and relative property T

2000 ◽  
Vol 20 (2) ◽  
pp. 483-499 ◽  
Author(s):  
PAUL JOLISSAINT
Keyword(s):  
Von Neumann Algebra ◽  
Approximation Property ◽  
Compact Groups ◽  
Approximation Properties ◽  
Von Neumann ◽  
Rigidity Results ◽  
Relative Property ◽  
Finite Von Neumann Algebra ◽  
Property T ◽  
Standard Probability

Let $G$ and $H$ be locally compact groups. Assume that $G$ acts on a standard probability space $(S,\mu)$, $\mu$ being $G$-invariant. We prove that if there exists a Borel cocycle $\alpha:S\times G\longrightarrow H$ which is proper in an appropriate sense, then $G$ inherits some approximation properties of $H$, for instance amenability or the so-called Haagerup Approximation Property. On the other hand, if $G_{0}$ is a closed subgroup of $G$, if the pair $(G,G_{0})$ has the relative property (T) of Margulis [19] and if either $H$ has Haagerup Approximation Property, or if it is the unitary group of a finite von Neumann algebra with a similar property, then we give rigidity results analogous to that in [23] and [1].

scholarly journals On the large-scale geometry of diffeomorphism groups of 1-manifolds

2018 ◽  
Vol 30 (1) ◽  
pp. 75-86
Author(s):  
Michael P. Cohen
Keyword(s):  
Banach Space ◽  
Large Scale ◽  
Local Property ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Relative Property ◽  
Large Scale Geometry ◽  
Isometry Class

Abstract We apply the framework of Rosendal to study the large-scale geometry of the topological groups {\operatorname{Diff}_{+}^{k}(M^{1})} , consisting of orientation-preserving {C^{k}} -diffeomorphisms (for {1\leq k\leq\infty} ) of a compact 1-manifold {M^{1}} ( {=I} or {\mathbb{S}^{1}} ). We characterize the relative property (OB) in such groups: {A\subseteq\operatorname{Diff}_{+}^{k}(M^{1})} has property (OB) relative to {\operatorname{Diff}_{+}^{k}(M^{1})} if and only if {\sup_{f\in A}\sup_{x\in M^{1}}\lvert\log f^{\prime}(x)|<\infty} and {\sup_{f\in A}\sup_{x\in M^{1}}|f^{(j)}(x)|<\infty} for every integer j with {2\leq j\leq k} . We deduce that {\operatorname{Diff}_{+}^{k}(M^{1})} has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if {k<\infty} . We show that the groups {\operatorname{Diff}_{+}^{1}(I)} and {\operatorname{Diff}_{+}^{1}(\mathbb{S}^{1})} are quasi-isometric to the infinite-dimensional Banach space {C[0,1]} .

scholarly journals Relative Inner Amenability and Relative Property Gamma

2016 ◽  
Vol 119 (2) ◽  
pp. 293
Author(s):  
Paul Jolissaint
Keyword(s):  
Discrete Group ◽  
Proper Subgroup ◽  
Type Ii ◽  
Semidirect Products ◽  
Relative Property ◽  
Property T ◽  
Equivalent Conditions ◽  
Inner Amenability

Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples coming mainly from semidirect products, as well as counter-examples. We also discuss the corresponding relative property gamma for pairs of type II$_1$ factors $N\subset M$ and we deduce from this a characterization of discrete, icc groups which do not have property (T).

scholarly journals A VON NEUMANN ALGEBRA CHARACTERIZATION OF PROPERTY (T) FOR GROUPOIDS

2018 ◽  
Vol 108 (3) ◽  
pp. 363-386
Author(s):  
MARTINO LUPINI
Keyword(s):  
Probability Measure ◽  
Von Neumann Algebra ◽  
Discrete Probability ◽  
Von Neumann ◽  
Relative Property ◽  
Property T ◽  
Algebra Characterization

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusions $L(H)\subset L(G)$.

scholarly journals Reduced 1-cohomology and relative property (T)

2010 ◽  
Vol 270 (3-4) ◽  
pp. 613-626 ◽  
Author(s):  
Talia Fernós ◽  
Alain Valette ◽  
Florian Martin
Keyword(s):  
Relative Property ◽  
Property T

scholarly journals Relative property (T) and linear groups

2006 ◽  
Vol 56 (6) ◽  
pp. 1767-1804 ◽  
Author(s):  
Talia Fernós
Keyword(s):  
Linear Groups ◽  
Relative Property ◽  
Property T
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