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

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

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

The casein kinases Yck1p and Yck2p act in the secretory pathway, in part, by regulating the Rab exchange factor Sec2p.

Sec2p is a guanine nucleotide exchange factor that activates Sec4p, the final Rab GTPase of the yeast secretory pathway. Sec2p is recruited to secretory vesicles by the upstream Rab Ypt32p acting in concert with phosphatidylinositol-4-phosphate (PI(4)P). Sec2p also binds to the Sec4p effector Sec15p, yet Ypt32p and Sec15p compete against each other for binding to Sec2p. We report here that the redundant casein kinases Yck1p and Yck2p phosphorylate sites within the Ypt32p/Sec15p binding region and in doing so promote binding to Sec15p and inhibit binding to Ypt32p. We show that Yck2p binds to the autoinhibitory domain of Sec2p, adjacent to the PI(4)P binding site, and that addition of PI(4)P inhibits Sec2p phosphorylation by Yck2p. Loss of Yck1p and Yck2p function leads to accumulation of an intracellular pool of the secreted glucanase Bgl2p, as well as to accumulation of Golgi-related structures in the cytoplasm. We propose that Sec2p is phosphorylated after it has been recruited to secretory vesicles and the level of PI(4)P has been reduced. This promotes Sec2p function by stimulating its interaction with Sec15p. Finally, Sec2p is dephosphorylated very late in the exocytic reaction to facilitate recycling

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