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Register a new userRapid decay and Metric Approximation Property
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Let Gamma be a discrete group satisfying the rapid decay property with respect to a length function which is conditionally negative. Then the reduced C*-algebra of Gamma has the metric approximation property. The central point of our proof is an observation that the proof of the same property for free groups due to Haagerup transfers directly to this more general situation. Examples of groups satisfying the hypotheses include free groups, surface groups, finitely generated Coxeter groups, right angled Artin groups and many small cancellation groups.
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This is a survey of methods of proving or disproving the Rapid Decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the property (**)-relative hyperbolicity from our paper with Cornelia Drutu, math/0405500, and implies the Rapid Decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, Artin groups of large type and the mapping class groups have the centroid property. We also present a simple non-amenability-like property that follows from RD, and give an easy example of a group without RD and without any amenable subgroup with superpolynomial growth, some misprints in other sections are corrected.
We study the geometry of infinitely presented groups satisfying the small cancelation condition C(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power continuum. The present work can be viewed as a first non-trivial step towards a systematic investigation of direct limits of hyperbolic groups.
We prove that the alternating group of a topologically free action of a countably infinite group $Gamma$ on the Cantor set has the property that all of its $ell^2$-Betti numbers vanish and, in the case that $Gamma$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Gamma$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austins result on the invariance of measure entropy under bounded orbit equivalence.
The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{rho}$ for each irreducible unitary $G$-representation $rho$, a relation $g_{rho}=g_{rho}g_{rho}$ whenever $rho$ is weakly contained in $rhootimes rho$, and $g_{rho^*}=g_{rho}^{-1}$ for the representation $rho^*$ contragredient to $rho$. $G$ satisfies chain-center duality if assigning to each $g_{rho}$ the central character of $rho$ is an isomorphism of $C(G)$ onto the dual $widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M{u}gers result compact groups satisfy chain-center duality.
We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, $omega$-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.
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