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1. Arnt, Sylvain. Large scale geometry and isometric affine actions on Banach spaces : Géométrie à grande échelle et actions isométriques affines sur des espaces de Banach.

Degree: Docteur es, Mathématiques, 2014, Université d'Orléans

Dans le premier chapitre, nous définissons la notion d’espaces à partitions pondérées qui généralise la structure d’espaces à murs mesurés et qui fournit un cadre géométrique à l’étude des actions isométriques affines sur des espaces de Banach pour les groupes localement compacts à base dénombrable. Dans un premier temps, nous caractérisons les actions isométriques affines propres sur des espaces de Banach en termes d’actions propres par automorphismes sur des espaces à partitions pondérées. Puis, nous nous intéressons aux structures de partitions pondérées naturelles pour les actions de certaines constructions de groupes : somme directe ; produit semi-directe ; produit en couronne et produit libre. Nous établissons ainsi des résultats de stabilité de la propriété PLp par ces constructions. Notamment, nous généralisons un résultat de Cornulier, Stalder et Valette de la façon suivante : le produit en couronne d’un groupe ayant la propriété PLp par un groupe ayant la propriété de Haagerup possède la propriété PLp. Dans le deuxième chapitre, nous nous intéressons aux espaces métriques quasi-médians - une généralisation des espaces hyperboliques à la Gromov et des espaces médians - et à leurs propriétés. Après l’étude de quelques exemples, nous démontrons qu’un espace δ-médian est δ′-médian pour tout δ′ ≥ δ. Ce résultat nous permet par la suite d’établir la stabilité par produit directe et par produit libre d’espaces métriques - notion que nous développons par la même occasion. Le troisième chapitre est consacré à la définition et l’étude d’une distance propre, invariante à gauche et qui engendre la topologie explicite sur les groupes localement compacts, compactement engendrés. Après avoir montré les propriétés précédentes, nous prouvons que cette distance est quasi-isométrique à la distance des mots sur le groupe et que la croissance du volume des boules est contrôlée exponentiellement.

In the first chapter, we define the notion of spaces with labelled partitions which generalizes the structure of spaces with measured walls : it provides a geometric setting to study isometric affine actions on Banach spaces of second countable locally compact groups. First, we characterise isometric affine actions on Banach spaces in terms of proper actions by automorphisms on spaces with labelled partitions. Then, we focus on natural structures of labelled partitions for actions of some group constructions : direct sum ; semi-direct product ; wreath product and free product. We establish stability results for property PLp by these constructions. Especially, we generalize a result of Cornulier, Stalder and Valette in the following way : the wreath product of a group having property PLp by a Haagerup group has property PLp. In the second chapter, we focus on the notion of quasi-median metric spaces - a generalization of both Gromov hyperbolic spaces and median spaces - and its properties. After the study of some examples, we show that a δ-median space is δ′-median for all δ′ ≥ δ. This result gives us a way to establish the stability…

Advisors/Committee Members: Chatterji, Indira Lara (thesis director).

Subjects/Keywords: Géométrie des groupes; Actions isométriques affines; Actions propres; Espaces de Banach; Espaces Lp; Groupes hyperboliques; Espaces à murs mesurés; Espaces médians; Propriété de Haagerup; Distances propres sur les groupes; Geometric group theory; Isometric affine actions; Proper actions; Banach spaces; Lp spaces; Hyperbolic groups; Spaces with measured walls; Median spaces; Haagerup property; Proper metrics on groups

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APA (6th Edition):

Arnt, S. (2014). Large scale geometry and isometric affine actions on Banach spaces : Géométrie à grande échelle et actions isométriques affines sur des espaces de Banach. (Doctoral Dissertation). Université d'Orléans. Retrieved from http://www.theses.fr/2014ORLE2021

Chicago Manual of Style (16th Edition):

Arnt, Sylvain. “Large scale geometry and isometric affine actions on Banach spaces : Géométrie à grande échelle et actions isométriques affines sur des espaces de Banach.” 2014. Doctoral Dissertation, Université d'Orléans. Accessed December 02, 2020. http://www.theses.fr/2014ORLE2021.

MLA Handbook (7th Edition):

Arnt, Sylvain. “Large scale geometry and isometric affine actions on Banach spaces : Géométrie à grande échelle et actions isométriques affines sur des espaces de Banach.” 2014. Web. 02 Dec 2020.

Vancouver:

Arnt S. Large scale geometry and isometric affine actions on Banach spaces : Géométrie à grande échelle et actions isométriques affines sur des espaces de Banach. [Internet] [Doctoral dissertation]. Université d'Orléans; 2014. [cited 2020 Dec 02]. Available from: http://www.theses.fr/2014ORLE2021.

Council of Science Editors:

Arnt S. Large scale geometry and isometric affine actions on Banach spaces : Géométrie à grande échelle et actions isométriques affines sur des espaces de Banach. [Doctoral Dissertation]. Université d'Orléans; 2014. Available from: http://www.theses.fr/2014ORLE2021

2. Pro, Curtis. On Riemannian Submersions and Diffeomorphism Stability.

Degree: Mathematics, 2012, University of California – Riverside

This thesis consists of work that was carried out in three separate papers that were written during my time at UC, Riverside. Abstract of chapter II: If π:M →  B is a Riemannian Submersion and M has non-negative sectional curvature, O'Neill's Horizontal Curvature Equation shows that B must also have non-negative curvature. We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.Abstract of Chapter III: Though Riemannian submersions preserve non-negative sectional curvature this does not generalize to Riemannian submersions from manifolds with non-negative Ricci curvature. We give here an example of a Riemannian submersion π: M →  B for which {Ricci}p(M)>0 and at some point p∈ B, {Ricci}p(B)<0. Abstract of Chapter IV: The smallest r so that a metric r – ball covers a metric space M is called the radius of M. The volume of a metric r-ball in the space form of constant curvature k is an upper bound for the volume of any Riemannian manifold with sectional curvature  ≥  k and radius  ≤  r. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.

Subjects/Keywords: Mathematics; Diffeomorphsim Stability; Dual Foliations; Isometric Group Actions; Ricci Curvature; Riemannian Submersions

…Problem 1 in the special case when the submersion is induced by an isometric group action with… …Since F is given by the orbit decomposition of an isometric group action, the dual foliation… …principal G-actions, hol(b) is always a Lie group. Recall (see [20], p… …of a Lie group with a bi-invariant metric ([9, 27, 48]). Earlier… …Riemannian submersion of a compact Lie group with a bi-invariant metric. Then 1 Every zero… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Pro, C. (2012). On Riemannian Submersions and Diffeomorphism Stability. (Thesis). University of California – Riverside. Retrieved from http://www.escholarship.org/uc/item/2z16d2kf

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Pro, Curtis. “On Riemannian Submersions and Diffeomorphism Stability.” 2012. Thesis, University of California – Riverside. Accessed December 02, 2020. http://www.escholarship.org/uc/item/2z16d2kf.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Pro, Curtis. “On Riemannian Submersions and Diffeomorphism Stability.” 2012. Web. 02 Dec 2020.

Vancouver:

Pro C. On Riemannian Submersions and Diffeomorphism Stability. [Internet] [Thesis]. University of California – Riverside; 2012. [cited 2020 Dec 02]. Available from: http://www.escholarship.org/uc/item/2z16d2kf.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Pro C. On Riemannian Submersions and Diffeomorphism Stability. [Thesis]. University of California – Riverside; 2012. Available from: http://www.escholarship.org/uc/item/2z16d2kf

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.