Infoscience

Thesis

Actions de relations d'équivalence sur les champs d'espaces métriques CAT(0)

This work is dedicated to the study of Borel equivalence relations acting on Borel fields of CAT(0) metric spaces over a standard probability space. In this new framework we get similar results to some theorems proved recently by S. Adams-W. Ballmann or N. Monod concerning groups of isometries of CAT(0) spaces. In Chapter 1, we build several Borel structures on a variety of fields before dealing in particular with Borel fields of CAT(0) spaces. Chapter 2 discusses the notion of an action for an equivalence relation on a field of metric spaces and gives several examples. We also introduce a definition of amenability for equivalence relations in terms of invariant section following an idea of R.J. Zimmer. Chapter 3 deals with the action of an amenable equivalence relation and shows that such a relation cannot act without fixing a section at infinity or preserving a subfield of Euclidean spaces. In Chapter 4, we show that if an equivalence relation is generated by two commuting groups and acts without fixing a section at infinity, then the field splits equivariantly and isometrically as a product. Using this result we also show that equivalence relations containing two coamenable subrelations cannot act without fixing a section at infinity or preserving a subfield of Euclidean spaces.

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