Yves de Cornulier

Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

Abstract.We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into a Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with “controlled” Følner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

Normal subgroups in the Cremona group

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane

On the isolated points in the space of groups

\mathbb{P}_{\mathbf{k}}^2

Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity

is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.

Finitely Presented Wreath Products and Double Coset Decompositions

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.

Strongly Bounded Groups and Infinite Powers of Finite Groups

We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups and solvability of the word problem.

METABELIAN GROUPS WITH QUADRATIC DEHN FUNCTION AND BAUMSLAG-SOLITAR GROUPS

Abstract We perform a systematic investigation of Kazhdans relative Property (T) for pairs ( G , X ) , where G is a locally compact group and X is any subset. When G is a connected Lie group or a p-adic algebraic group, we provide an explicit characterization of subsets X ⊂ G such that ( G , X ) has relative Property (T). In order to extend this characterization to lattices Γ ⊂ G , a notion of “resolutions” is introduced, and various characterizations of it are given. Special attention is paid to subgroups of SU ( 2 , 1 ) and SO ( 4 , 1 ) .

Contracting automorphisms and Lp-cohomology in degree one

Our main result is that the simple Lie group G = Sp(n, 1) acts metrically properly isometrically on Lp(G) if p > 4n + 2. To prove this, we introduce Property