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

Abstract

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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Related concepts (14)

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equiva

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function.

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection,

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.

Martin Anderegg, Philippe Paul Antoine Henry

We present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

The field of computational topology has developed many powerful tools to describe the shape of data, offering an alternative point of view from classical statistics. This results in a variety of complex structures that are not always directly amenable for machine learning tasks. We develop theory and algorithms to produce computable representations of simplicial or cell complexes, potentially equipped with additional information such as signals and multifiltrations. The common goal of the topics discussed in this thesis is to find reduced representations of these often high dimensional and complex structures to better visualize, transform or formulate theoretical results about them. We extend the well known graph learning algorithm node2vec to simplicial complexes, a higher dimensional analogue of graphs. To this end we propose a way to define random walks on simplicial complexes, which we then use to design an extension of node2vec called k-simplex2vec, producing a representation of the simplices in a Euclidean space. Furthermore, the study of this method leads to interesting questions about robustness of graph and simplicial learning methods. In the case of graphs, we study node2vec embeddings arising from different parameter sets, analysing their quality and stability using various measures. In the topic of signal processing, we explore how discrete Morse theory can be used for compression and reconstruction of cell complexes equipped with signals. In particular we study the effect of the compression of a complex on the Hodge decomposition of its signals. We study how the signal changes through compression and reconstruction by introducing a topological reconstruction error, showing in particular that part of the Hodge decomposition is preserved. Moreover, we prove that any deformation retract over R can be expressed as a Morse deformation retract in a well-chosen basis, thus extending the reconstruction results to any deformation retract. In addition, we introduce an algorithm to minimize the loss induced by the reconstruction of a compressed signal. Finally, we use discrete Morse theory to compute an invariant of multi-parameter persistent homology, the rank invariant. We can restrict a multi-parameter persistence module to a one- dimensional persistence module along any line of positive slope and compute the one-dimensional analogue of the rank invariant, namely the barcode. Through a discrete Morse matching we can determine critical values in the multifiltration, which in turn allows us to identify equivalence classes of lines in the parameter space. In our main result, we explain how to compute the barcode along any given line of an equivalence class given the barcode along a representative line. This provides a way to fiber the rank invariant according to the critical values of a discrete Morse matching and to perform computations in the corresponding one-dimensional module, which is much better understood.