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

Auteur(s)
Dreesen, Dennis
Editeur(s)
Date de parution
2011
Mots-clés
Résumé
A crystallographic group is a group that acts faithfully, isometrically and properly discontinuously on a Euclidean space R<sup>n</sup> 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 R<sup>n</sup>, we consider actions on products <i>M</i> x <i>N</i> where <i>N</i> is a simply connected, connected nilpotent Lie-group equipped with a left-invariant Riemannian metric and where <i>M</i> is a closed Riemannian manifold. Our proof to generalize the first Bieberbach theorem to this setting, needs that the isometries of <i>M</i> x <i>N</i> split, i.e that Iso(<i>M</i> x <i>N</i>) = Iso(<i>M</i>) x Iso(<i>N</i>). In Part I of this thesis, we introduce a class of products on which the isometries split. <br> Consequently, going back to the Bierbach context, we can replace Euclidean space R<sup>n</sup> 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. <br> 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.
Notes
Thèse de doctorat : Université de Neuchâtel, 2011 ; 2191
Identifiants
https://libra.unine.ch/handle/123456789/11997
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10.35662/unine-thesis-2191
Type de publication
doctoral thesis
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