As Avez showed (in 1970), the fundamental group of a compact Riemannian manifold of nonpositive sectional curvature has exponential growth if and only if it is not flat. After several generalizations from Gromov, Zimmer, Anderson, Burger and Shroeder, the following theorem was proved by Adams and Ballmann (in 1998). Theorem Let X be a proper CAT(0) space. If Γ is an amenable group of isometries of X, then at least one of the following two assertions holds: Γ fixes a point in ∂X (boundary of X). X contains a Γ-invariant flat (isometric copy of Rn, n ≥ 0). Following an idea of my PhD advisor Nicolas Monod, I tried to generalize this theorem in the context of goupoids, in this case Borel G-spaces and countable Borel equivalence relations. This lead me to study the notion of Borel fields of metric spaces, which turns out to be a suitable context to define an action of a countable Borel equivalence relation. A field of metric spaces over a set Ω is a family {(Xω,dω)} ω∈Ω of nonempty metric spaces denoted by (Ω,X•). We introduced as S( Ω,X•) the set of maps Such maps are called sections. If Ω is a Borel space, we can define a Borel structure on a field of metric spaces to be a subset Lℒ( Ω,X•) of S( Ω,X•) satisfying these three conditions For all f, g ∈ ℒ(Ω,X•), the function Ω → R, ω → dω(f(ω), g(ω)) is Borel. If h ∈ S(Ω,X•) is such that the function Ω → R, ω → dω(f(ω), h(ω)) is Borel for all f ∈ ℒ(Ω,X•), then h ∈ ℒ(Ω,X•). There exists a countable family of sections {fn}n≥1 ⊆ ℒ(Ω,X•) such that {fn (ω)}n≥1 = Xω for all ω ∈ Ω. This definition is consistent with more classical definitions of Borel fields of Banach spaces or of Borel fields of Hilbert spaces. The notion of a Borel field of metric spaces has been used in convex analysis and in economy. As said before, we can define an action of a countable Borel equivalence relation ℛ ⊆ Ω2 on a Borel field of metric spaces (Ω,X•) in a natural way. It's determined by a family of bijectives maps {α(ω, ω') : Xω → Xω'}(ω,ω')∈ℛ such that For all (ω,ω'), (ω',ω") ∈ ℛ the following equality is satisfied α(ω', ω") ◦ α(ω, ω') = α(ω, ω"). For all f, g ∈ ℒ(Ω,X), the function ℛ → R, (ω, ω') → dω(f(ω), α(ω', ω)g(ω')) is Borel. Zimmer (1977) introduced the notion of amenability for ergodic G-spaces and equivalence relations, of which we obtained the first generalization (in collaboration with Philippe Henry). Theorem Let R be a countable, Borel, preserving the class of the measure, ergodic and amenable equivalence relation on the probability space Ω acting on a Borel field ( Ω,X•) of proper CAT(0) spaces with finite topological dimension. Then at least one of the following assertions is true: There exists an ℛ-invariant Borel section ξ ∈ L(Ω,∂X•). There exists an ℛ-invariant Borel subfield (Ω, F•) of (Ω,X•) consisting of flat subsets. And the second generalization for amenable ergodic G-spaces. Theorem Let G be a locally compact second countable group, Ω a preserving class of the measure, ergodic amenable G-space, X a proper CAT(0) space with finite topological dimension and α : G × Ω → Iso(X) a Borel cocycle. Then at least one of the following assertions is true: There exists an α-invariant Borel function ξ : Ω → ∂X. There exists an α-invariant borelian subfield (Ω, F•) of the trivial field (Ω, X) consisting of flat subsets. If we consider (Ω,μ) to be a strong boundary of the group G, the cocycle α to come from an action of G on X, and X to have flats of at most dimension 2, then we can conclude the following. Theorem Let G be a locally compact second countable group, (Ω,μ) a strong boundary of G, X a proper CAT(0) space with finite topological dimension and whose flats are of dimension at most 2. Let suppose that G acts by isometry on X. Then at least one of the following assertions is true: There exists a G-equivariant Borel function ξ: Ω → ∂X. There exists a G-invariant flat F in X. The proof of the three theorems are strongly based on properties of Borel field of metric spaces that we prove in this thesis.
EPFL2010
