Fixed point properties in the space of marked groups

We explain, following Gromov, how to produce uniform isometric actions of groups starting from isometric actions without fixed point, using common ultralimits techniques. This gives in particular a simple proof of a result by Shalom: Kazhdan's property (T) defines an open subset in the space of marked finitely generated groups.Comment: The only modification from previous version is section numbering, in order to agree with the published versio

Proper actions of wreath products and generalizations

We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also allows for a characterization of subsets with relative Property T in a standard wreath product.Comment: 29 pages, Minor change

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

Strongly singular MASA's and mixing actions in finite von Neumann algebras

Let $\Gamma$ be a countable group and let $\Gamma_0$ be an infinite abelian subgroup of $\Gamma$. We prove that if the pair $(\Gamma,\Gamma_0)$ satisfies some combinatorial condition called (SS), then the abelian subalgebra $A=L(\Gamma_0)$ is a singular MASA in $M=L(\Gamma)$ which satisfies a weakly mixing condition. If moreover it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, HNN extentions and semidirect products, and in particular we exhibit examples of singular MASA's that satisfy the weak mixing condition but not the strong mixing one.Comment: Title updated, examples and references added. To appear in Ergod. Th. & Dynam. Sys

Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups

We prove that the limits of Baumslag-Solitar groups which we previously studied are non-linear hopfian C*-simple groups with infinitely many twisted conjugacy classes. We exhibit infinite presentations for these groups, classify them up to group isomorphism, describe their automorphisms and discuss the word and conjugacy problems. Finally, we prove that the set of these groups has non-zero Hausforff dimension in the space of marked groups on two generators.Comment: 30 pages, no figures, englis

Proper actions of lamplighter groups associated with free groups

Given a finite group $H$ and a free group $F_n$, we prove that the wreath product $H\wr F_n$ admits a metrically proper, isometric action on a Hilbert space.Comment: 6 pages. The part on Hilbert space compression from the first version of this paper, will be incorporated into a more elaborate paper on the subjec