338 research outputs found
Wreath products with the integers, proper actions and Hilbert space compression
We prove that the properties of acting metrically properly on some space with
walls or some CAT(0) cube complex are closed by taking the wreath product with
\Z. We also give a lower bound for the (equivariant) Hilbert space compression
of H\wr\Z in terms of the (equivariant) Hilbert space compression of H.Comment: Minor correction
L^2-Betti numbers and Plancherel measure
We compute -Betti numbers of postliminal, locally compact, unimodular
groups in terms of ordinary dimensions of reduced cohomology with coefficients
in irreducible unitary representations and the Plancherel measure. This allows
us to compute the -Betti numbers for semi-simple Lie groups with finite
center, simple algebraic groups over local fields, and automorphism groups of
locally finite trees acting transitively on the boundary.Comment: 11 page
Unbounded symmetric operators in -homology and the Baum-Connes Conjecture
Using the unbounded picture of analytical K-homology, we associate a
well-defined K-homology class to an unbounded symmetric operator satisfying
certain mild technical conditions. We also establish an ``addition formula''
for the Dirac operator on the circle and for the Dolbeault operator on closed
surfaces. Two proofs are provided, one using topology and the other one,
surprisingly involved, sticking to analysis, on the basis of the previous
result. As a second application, we construct, in a purely analytical language,
various homomorphisms linking the homology of a group in low degree, the
K-homology of its classifying space and the analytic K-theory of its
C^*-algebra, in close connection with the Baum-Connes assembly map. For groups
classified by a 2-complex, this allows to reformulate the Baum-Connes
Conjecture.Comment: 42 pages, 3 figure
On Godement's characterisation of amenability
Motivated by a question related to the construction of the Baum-Connes analytical assembly map for locally compact groups, we refine a criterion of Godement for amenability: for a unimodular group G, our criterion says that G is amenable if and only if every compactly supported, positive-definite function has non-negative integral over
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