Equivariant and non-equivariant uniform embeddings into products and Hilbert spaces

Abstract

A crystallographic group is a group that acts faithfully, isometrically and properly discontinuously on a Euclidean space Rn and the theory of crystallographic groups is in some sense governed by three main theorems, called the Bieberbach theorems. The research performed in this thesis is motivated from a desire to generalize these theorems to a more general setting. First, instead of actions on Rn, we consider actions on products M x N where N is a simply connected, connected nilpotent Lie-group equipped with a left-invariant Riemannian metric and where M is a closed Riemannian manifold. Our proof to generalize the first Bieberbach theorem to this setting, needs that the isometries of M x N split, i.e that Iso(M x N) = Iso(M) x Iso(N). In Part I of this thesis, we introduce a class of products on which the isometries split. Consequently, going back to the Bierbach context, we can replace Euclidean space Rn by the class of all, possibly infinite-dimensional, Hilbert spaces. We here enter the world of groups with the Haagerup property. Quantifying the degree to which a group satisfies the Haagerup property leads to the notion of equivariant Hilbert space compression, and we investigate the behaviour of this number under group constructions in Part II. Finally, dropping the condition that groups under consideration must act isometrically on a Hilbert space, we look, in part III, at mere (uniform) embeddings of groups into Hilbert spaces. Quantifying the degree to which a group embeds uniformly into a Hilbert space, leads to the notion of (ordinary) Hilbert space compression and in Part III, the behaviour of this number under group constructions is investigated.