A group W is said to have property FA_n if every action of W by isometries on an n-dimensional CAT(0) cell complex has a global fixed point. We construct a complex on which a Coxeter group W acts by cellular isometries without global fixed points, and show that under certain combinatorial conditions, complexes constructed in this way are CAT(0). We then construct several infinite classes of Coxeter groups which have property FA_2 but not property FA_3. / text

We calculate the second and third integral homology of arbitrary finite rank Coxeter groups. The first of these calculations refines a theorem of Howlett, the second is entirely new. We then prove that families of Artin monoids, which have the braid monoid as a submonoid, satisfy homological stability. When the K(π,1) conjecture holds this gives a homological stability result for the associated families of Artin groups. In particular, we recover a classic result of Arnol'd.

This mathematics thesis deals with combinatorial representation theory. Cells were introduced in a 1979 paper written by D. Kazhdan and G. Lusztig, and have intricate links with many areas of mathematics, including the representation theory of Coxeter groups, Iwahori–Hecke algebras, semisimple complex Lie algebras, reductive algebraic groups and Lie groups. One of the main problems in the theory of cells is their classification for all finite Coxeter groups. This thesis is a detailed study of cells in type Bn with respect to certain choices of parameters, and contributes to the classification by giving the first characterisation of left cells when b/a = n − 1. Other results include the introduction of a generalised version of the enhanced right descent set and exhibiting the asymptotic left cells of type Bn as left Vogan classes. Combinatorial results give rise to efficient algorithms so that cells can be determined with a computer; the methods involved in this work transfer to a new, faster way of calculating the cells with respect to the studied parameters. The appendix is a Python file containing code to make such calculations.

Originally presented as the author's Thesis (Ph. D.)--Scuola normale superiore Pisa. / Includes bibliographical references (p. [125]-131) and index.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 51-52).

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O presente trabalho tem por objeto fazer uma introdução ao estudo das subvariedades isoparamétricas do espaço Euclidiano. Começamos com uma introdução ao desenvolvimento histórico desses objetos. A seguir apresentamos os conceitos básicos da teoria de subvariedades de formas espaciais. Deduzimos as equações fundamentais de primeira e segunda ordem e demonstramos o teorema fundamental da teoria de subvariedades. Em seguida damos a definição de subvariedade isoparamétrica e desenvolvemos conceitos elementares para o caso do espaço Euclidiano como são normais de curvatura, grupo de Coxeter, câmera de Weyl e variedades paralelas e focais. Provamos dois teoremas referentes à decomposição de subvariedades isoparamétricas do espaço Euclidiano adaptando ferramentas usadas em [HL97] para ocaso de subvariedades isoparamétricas de espaços de Hilbert. Demonstramos o teorema da fatia e discutimos sobre subvariedades isoparamétricas desde o ponto de vista clássico, a saber, aplicações isoparamétricas. Concluímos com alguns exemplos: hipersuperfécies isoparamétricas da esfera e órbitas principais da ação adjunta de um grupo de Lie sobre a respectiva álgebra de Lie. / The goal of this dissertation is to present an introduction to the study of isoparametric submanifolds of Euclidean space. We begin with an introduction to the history of the subject. Then we present the basic results of submanifold theory of space forms. We compute the fundamental equations of first and second order, and we prove the fundamental theorem of submanifold theory. Next, we define isoparametric submanifolds and discuss some basic constructions, as curvature normals, Coxeter groups, Weyl chambers and parallel and focal submanifolds. We prove two decomposition theorems about isoprametric submanifolds using techniques that we learnt from [HL97], paper in which the case of submanifolds of Hilbert spaces is studied. Then we prove slice theorem. We also discuss those submanifold from the classical point of view, namely, isoparametric maps. We finish by explaining some examples: isoparametric hipersurfaces of spheres and principal orbits of the adjoint action of a Lie group on its Lie algebra.

Cette thèse se propose d'étudier sous divers points de vue les groupes d'automorphismes d'immeubles. Un de ses objectifs est de mettre en valeur les différences autant que les analogies entre les immeubles affines et non affines. Pour appuyer cette dichotomie, on y démontre que les groupes d'automorphismes d'immeubles non affines n'ont jamais de paire de Gelfand, contrairement aux immeubles affines. Dans l'autre sens, pour souligner l'analogie entre immeubles affines et non affines, on définit une nouvelle notion de bord combinatoire d'un immeuble. Dans le cas des immeubles affines, ce bord s'identifie au bord polyédral. On relie la construction de ce bord à d'autres constructions déjà existantes, par exemple, la compactification de Busemann du graphe des chambres. La compactification combinatoire est également isomorphe à la compactification par la topologie de Chabauty de l'ensemble des chambres, sous des hypothèses de transitivité. On relie aussi le bord combinatoire à un autre espace, généralisant une construction de F. Karpelevic pour les espaces symétriques : celle du bord raffiné d'un espace CAT(0).On démontre alors que les points du bord paramètrent les sous-groupes moyennables maximaux de l'immeuble, à indice fini près. Enfin, on prouve que l'action du groupe d'automorphismes d'un immeuble localement fini sur le bord combinatoire de ce dernier est moyennable, fournissant ainsi des résolutions en cohomologie bornée et des applications bord explicites. Ceci donne aussi une nouvelle preuve que ces groupes satisfont la conjecture de Novikov. / The object of this thesis is the study, from different point of views, of automorphism groups of buildings. One of its objectives is to highlight the differences as well as the analogies between affine and non-affine buildings. In order to support this dichotomy, we prove that automorphism groups of non-affine buildings never have a Gelfand pair, contrarily to affine buildings.In the other direction, the analogy between affine and non-affine buildings is supported by the new construction of a combinatorial boundary of a building. In the affine case, this boundary is in fact the polyhedral boundary. We connect the construction of this boundary to other compactifications, such as the Busemann compactification of the graph of chambers. The combinatorial compactification is also isomorphic to the group-theoretic compactification, which embeds the set of chambers into the set of closed subgroups of the automorphism group. We also connect the combinatorial boundary to another space, which generalises a construction of F. Karpelevic for symmetric spaces : the refined boundary of a CAT(0) space.We prove that the maximal amenable subgroups of the automorphism group are, up to finite index, parametrised by the points of the boundary. Finally, we prove that the action of the automorphism group of a locally finite building on its combinatorial boundary is amenable, thus providing resolutions in bounded cohomology and boundary maps. This also gives a new proof that these groups satisfy the Novikov conjecture.

O presente trabalho tem por objeto fazer uma introdução ao estudo das subvariedades isoparamétricas do espaço Euclidiano. Começamos com uma introdução ao desenvolvimento histórico desses objetos. A seguir apresentamos os conceitos básicos da teoria de subvariedades de formas espaciais. Deduzimos as equações fundamentais de primeira e segunda ordem e demonstramos o teorema fundamental da teoria de subvariedades. Em seguida damos a definição de subvariedade isoparamétrica e desenvolvemos conceitos elementares para o caso do espaço Euclidiano como são normais de curvatura, grupo de Coxeter, câmera de Weyl e variedades paralelas e focais. Provamos dois teoremas referentes à decomposição de subvariedades isoparamétricas do espaço Euclidiano adaptando ferramentas usadas em [HL97] para ocaso de subvariedades isoparamétricas de espaços de Hilbert. Demonstramos o teorema da fatia e discutimos sobre subvariedades isoparamétricas desde o ponto de vista clássico, a saber, aplicações isoparamétricas. Concluímos com alguns exemplos: hipersuperfécies isoparamétricas da esfera e órbitas principais da ação adjunta de um grupo de Lie sobre a respectiva álgebra de Lie. / The goal of this dissertation is to present an introduction to the study of isoparametric submanifolds of Euclidean space. We begin with an introduction to the history of the subject. Then we present the basic results of submanifold theory of space forms. We compute the fundamental equations of first and second order, and we prove the fundamental theorem of submanifold theory. Next, we define isoparametric submanifolds and discuss some basic constructions, as curvature normals, Coxeter groups, Weyl chambers and parallel and focal submanifolds. We prove two decomposition theorems about isoprametric submanifolds using techniques that we learnt from [HL97], paper in which the case of submanifolds of Hilbert spaces is studied. Then we prove slice theorem. We also discuss those submanifold from the classical point of view, namely, isoparametric maps. We finish by explaining some examples: isoparametric hipersurfaces of spheres and principal orbits of the adjoint action of a Lie group on its Lie algebra.