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My dissertation focuses on the existence of metrics of non-positive curvature for the simplicial complexes constructed recently by Tom Brady and Daan Krammer for the braid groups and other Artin groups of finite type. In particular, for each Artin group of finite type, there is a recently constructed finite simplicial Eilenberg-Mac Lane space known as its Brady-Krammer complex. The Brady-Krammer complexes are highly symmetric objects. Prior work on the relationship between the Brady-Krammer complexes and the theory of CAT(0)spaces has produced some positive results in low-dimensions. More specifically, the Brady-Krammer complexes of dimension at most 3 have been shown to support piecewise Euclidean metrics of non-positive curvature. Similarly, the 4dimensional Brady-Krammer complexes of type A4 and type B4 also support such metrics. In every instance, the metrics assigned respect all of the symmetries alluded to above. The main results of my dissertation show that this pattern does not extend to the Brady-Krammer complexes of type F4 and D4. These are the first negative results known about the curvature of these Brady-Krammer complexes. The proofs of my main theorems involve a combination of combinatorial results and computer calculations. These negative results are particularly striking since Ruth Charney, John Meier and Kim Whittlesey have shown that a particular complex closely related to each Brady-Krammer complex admits an asymmetric metric satisfying a weak version of non-positive curvature. Thus, one corollary of my results is that the weak asymmetric version of a CAT(0) metric (initially defined by Mladen Bestvina) is strictly weaker than the traditional version.

My dissertation focuses on the existence of metrics of non-positive curvature for the simplicial complexes constructed recently by Tom Brady and Daan Krammer for the braid groups and other Artin groups of finite type. In particular, for each Artin group of finite type, there is a recently constructed finite simplicial Eilenberg-Mac Lane space known as its Brady-Krammer complex. The Brady-Krammer complexes are highly symmetric objects. Prior work on the relationship between the Brady-Krammer complexes and the theory of CAT(0)spaces has produced some positive results in low-dimensions. More specifically, the Brady-Krammer complexes of dimension at most 3 have been shown to support piecewise Euclidean metrics of non-positive curvature. Similarly, the 4dimensional Brady-Krammer complexes of type A4 and type B4 also support such metrics. In every instance, the metrics assigned respect all of the symmetries alluded to above. The main results of my dissertation show that this pattern does not extend to the Brady-Krammer complexes of type F4 and D4. These are the first negative results known about the curvature of these Brady-Krammer complexes. The proofs of my main theorems involve a combination of combinatorial results and computer calculations. These negative results are particularly striking since Ruth Charney, John Meier and Kim Whittlesey have shown that a particular complex closely related to each Brady-Krammer complex admits an asymmetric metric satisfying a weak version of non-positive curvature. Thus, one corollary of my results is that the weak asymmetric version of a CAT(0) metric (initially defined by Mladen Bestvina) is strictly weaker than the traditional version.

La classe des graphes quasi-médians est une généralisation des graphes médians, ou de manière équivalente, des complexes cubiques CAT(0). L'objectif de cette thèse est d'introduire ces graphes dans le monde de la théorie géométrique des groupes. Dans un premier temps, nous étendons la notion d'hyperplan définie dans les complexes cubiques CAT(0), et nous montrons que la géométrie d'un graphe quasi-médian se réduit essentiellement à la combinatoire de ses hyperplans. Dans la deuxième partie de notre texte, qui est le cœur de la thèse, nous exploitons la structure particulière des hyperplans pour démontrer des résultats de combinaison. L'idée principale est que si un groupe agit d'une bonne manière sur un graphe quasi-médian de sorte que les stabilisateurs de cliques satisfont une certaine propriété P de courbure négative ou nulle, alors le groupe tout entier doit satisfaire P également. Les propriétés que nous considérons incluent : l'hyperbolicité (éventuellement relative), les compressions lp (équivariantes), la géométrie CAT(0) et la géométrie cubique. Finalement, la troisième et dernière partie de la thèse est consacrée à l'application des critères généraux démontrés précédemment à certaines classes de groupes particulières, incluant les produits graphés, les groupes de diagrammes introduits par Guba et Sapir, certains produits en couronne, et certains graphes de groupes. Les produits graphés constituent notre application la plus naturelle, où le lien entre le groupe et son graphe quasi-médian associé est particulièrement fort et explicite; en particulier, nous sommes capables de déterminer précisément quand un produit graphé est relativement hyperbolique. / The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its hyperplanes. In the second part, we exploit the specific structure of the hyperplanes to state combination results. The main idea is that if a group acts in a suitable way on a quasi-median graph so that clique-stabilisers satisfy some non-positively curved property P, then the whole group must satisfy P as well. The properties we are interested in are mainly (relative) hyperbolicity, (equivariant) lp-compressions, CAT(0)-ness and cubicality. In the third part, we apply our general criteria to several classes of groups, including graph products, Guba and Sapir's diagram products, some wreath products, and some graphs of groups. Graph products are our most natural examples, where the link between the group and its quasi-median graph is particularly strong and explicit; in particular, we are able to determine precisely when a graph product is relatively hyperbolic.