Marches al\'eatoires et \'el\'ements contractants sur des espaces CAT(0)

This thesis is dedicated to random walks on spaces with non-positive curvature. In particular, we study the case of group actions on CAT(0) spaces that admit contracting elements, that is, whose properties mimic those of loxodromic isometries in Gromov-hyperbolic spaces. In this context, we prove several limit laws, among which the almost sure convergence to the boundary without moment assumption, positivity of the drift and a central limit theorem. In a second part, we study boundary maps and stationary measures on affine buildings of type $\tilde{A}_2$, and we show that there always exists a hyperbolic isometry for a non-elementary action by isometries on such a space. Our approach involves the use of hyperbolic models for CAT(0) spaces, which were constructed by H.~Petyt, D.~Spriano and A.~Zalloum, and measured boundary theory, whose principles come from H.~Furstenberg.Comment: 194 pages. Introduction and presentation of the context in Frenc

Moyennabilité et courbure:G-espaces boréliens et espaces CAT(0)

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

Espaces de repr\'esentations compl\`etement r\'eductibles

We study some geometric properties of actions on nonpositively curved spaces related to complete reducibility and semisimplicity, focusing on representations of a finitely generated group in the group G of rational points of a reductive group over a local field, acting on the associated space (symmetric space or affine building). We prove that the space of completely reducible classes is the maximal Hausdorff quotient space for the conjugacy action of G on the representation space.Comment: 15 page

Morphismes injectifs entre groupes d'Artin-Tits

We construct a family of morphisms between Artin-Tits groups which generalise the ones constructed by J. Crisp in [Injective maps between Artin groups, Proceedings of the Special Year in Geometric Group Theory, Berlin, (1999), 119 -- 138]. We show that their restrictions to the positive Artin monoids respect normal forms, and that for Artin-Tits groups of type FC, these morphisms are injective. The proof of the second result uses the Deligne Complex, and the normal cube paths constructed in [G. Niblo and L. Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621-633] and [J.A. Altobelli and R. Charney, A geometric Rational Form for Artin Groups of FC type, Geom. Dedicata, 79 (2000) 277-289]. Resume: On construit une classe de morphismes entre groupes d'Artin-Tits qui generalise celle construite par J. Crisp dans [Injective maps between Artin groups, Proceedings of the Special Year in Geometric Group Theory, Berlin, (1999), 119 -- 138]. On montre que leurs restrictions aux monoides respectent les formes normales, et que pour les groupes d'Artin-Tits de type FC ces morphismes sont injectifs. La demonstration du second resultat utilise le complexe de Deligne et les chemins cubiques normaux construits dans [G. Niblo et L. Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621-633] et [J.A. Altobelli et R. Charney, A geometric Rational Form for Artin Groups of FC type, Geom. Dedicata, 79 (2000) 277-289].Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-25.abs.htm