DAN

Abstract

If $(X,d)$ is a metric space, the Hilbert compression of $X$ is the supremum of all $\alpha$'s for which there exists a Lipschitz embedding $f$ from X to a Hilbert space, such that $C.d(x,y)^\alpha \leq \|f(x)-f(y)\|$ for every $x,y\in X$. When $G$ is a finitely generated group, Hilbert compression is a quasi-isometry invariant which has been related to concepts such as exactness, amenability, Haagerup property. In this survey talk, we will review the known results about the range of this invariant, then we will move on to some recent results (due to Naor-Peres, Li, Dreesen) on the behaviour of Hilbert compression under various group constructions (wreath products, free and amalgamated products, HNN-extensions, etc...).