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.