Valette, Alain

2007, De Cornulier, Yves, Tessera, Romain, Valette, Alain

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" Folner 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.

1993, Bekka, Bachir, Valette, Alain

1997, Bekka, Bachir, Valette, Alain

For an infinite, finitely generated group Gamma, we study the first cohomology group H-1(Gamma, lambda(Gamma)) with coefficients in the left regular representation lambda(Gamma) of Gamma on l(2)(Gamma). We first prove that H-1(Gamma, C Gamma) embeds into H-1(Gamma, lambda(Gamma)); as a consequence, if H-1(Gamma, lambda(Gamma)) = 0, then Gamma is not amenable with one end. For a Cayley graph X of Gamma, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Gamma is not amenable, then there is a natural isomorphism between H-1(Gamma, lambda(Gamma)) and HD(X)/C (the latter space being isomorphic to the first L-2-cohomology space of Gamma). We draw the following consequences: (1) If Gamma has infinitely many ends, then HD(X) not equal C; (2) If Gamma has Kazhdan's property (T), then HD(X) = C; (3) The property H-1(Gamma, lambda(Gamma)) = 0 is a quasi-isometry invariant; (4) Either H-1(Gamma, lambda(Gamma)) = 0 or H-1(Gamma, lambda(Gamma)) is infinite-dimensional; (5) If Gamma = Gamma(1) x Gamma(2) with Gamma(1) non-amenable and Gamma(2) infinite, then H-1(Gamma, lambda(Gamma)) = 0.

1997, Valette, Alain

Let U(M) be the unitary group of a finite, injective von Neumann algebra M. We observe that any subrepresentation of a group representation into U(M) is amenable in the sense of Bekka; this yields short proofs of two known results-one by Robertson, one by Kaagerup-concerning group representations into U(M).