In this thesis we describe how to estimate the distance spanned in the pants graph by a train track splitting sequence on a surface, up to multiplicative and additive constants. If some moderate assumptions on a splitting sequence are satisfied, each vertex set of a train track in it will represent a vertex of a graph which is naturally quasi-isometric to the pants graph; moreover the splitting sequence gives an edge-path in this graph so, more precisely, our distance estimate holds between the extreme points of this path. The present distance estimate is inspired by a result of Masur, Mosher and Schleimer for distances in the marking graph. However, we can apply their line of proof only after some manipulation of the splitting sequence: a rearrangement, changing the order the elementary moves are performed in, so that the ones producing Dehn twists are brought together; and then an untwisting, which suppresses the majority of these latter moves to give a new sequence, which does not end with the same track as before, but does not include any portion that is almost stationary in the pants graph. The required distance is then, up to constants, the number of splits occurring in the untwisted sequence. A consequence of our main theorem together with a result of Brock is that, given a pseudo-Anosov self-diffeomorphism ψ of a surface S, the maximal splitting sequence introduced by Agol gives us an estimate for the hyperbolic volume of the mapping torus built from S and ψ. There are also some interesting consequences for the hyperbolic volume of a solid torus minus a closed braid, via a machinery employed by Dynnikov and Wiest.

Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n−2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n−1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the generalization of Adam's conjecture to tunnel number n ≥ 2 manifolds may be false.

Cette thèse est une contribution au domaine des cubulations de groupes hyperboliques au sens de Gromov. Nous nous intéressons au cas particulier des groupes fondamentaux de variétés hyperboliques réelles compactes. La philosophie inspirée dans ce domaine par les travaux de M. Sageev est que si un groupe hyperbolique possède suffisamment de sous-groupes de codimension 1 quasi-convexes, alors il agit géométriquement sur un complexe cubique CAT(0) de dimension finie. Nous démontrons un critère précis de cubulation pour les groupes fondamentaux de variétés hyperboliques compactes, à l'aide de constructions d'espaces à murs quasi-isométriques à l'espace hyperbolique réel. Nous nous restreignons par la suite au cas particulier de la dimension 3 et plus particulièrement aux 3-variétés hyperboliques compactes virtuellement fibrées sur le cercle. Nous exploitons alors une construction de surfaces immergées incompressibles dites coupées-croisées due à D. Cooper, D. Long et A. Reid dans une telle 3-variété M pour fabriquer des sous-groupes de surface de son groupe fondamental~G. En raffinant des arguments de J. Masters et en exploitant la structure de l'application de Cannon-Thurston, nous parvenons à construire des sous-groupes de surfaces quasi-convexes de G en quantité suffisante pour que leurs ensembles limites permettent de séparer toutes les paires de points distincts du bord du revêtement universel de M. En conséquence de cette construction, G agit géométriquement sur un complexe cubique CAT(0) de dimension finie. D. Wise soulève alors la question de savoir si ce groupe G peut agir géométriquement et également virtuellement co-spécialement (au sens de F. Haglund et D. Wise) sur un complexe cubique CAT(0). Une réponse positive résoudrait les conjectures selon lesquelles G est large et le premier nombre de Betti virtuel de M est infini. Nous faisons remarquer que pour obtenir une réponse positive à cette question, il suffit de trouver une surface coupée-croisée virtuellement plongée dans un revêtement fini fibré sur le cercle de M. Nous concluons en présentant des conditions algébriques, puis géométriques et cohomologiques suffisantes pour qu'une surface coupée-croisée donnée soit virtuellement plongée. / This thesis contributes to the study of geometric actions of word-hyperbolic groups on finite dimensional CAT(0) cube complexes. We are mainly interested in the case of fundamental groups of closed hyperbolic manifolds. The philosophy coming from pioneer work of M. Sageev is that a hyperbolic group with sufficiently many quasi-convex codimension one subgroups acts geometrically on a finite dimensional CAT(0) cube complex. We prove a precise criterion for cubulation in the case of closed hyperbolic manifolds, by constructing spaces with walls quasi-isometric to real hyperbolic space. We next focus on the case of three dimensional closed hyperbolic manifolds which are virtually fibered over the circle. In this setting, we use a construction of incompressibly immersed cut-and-cross-join surfaces due to D. Cooper, D. Long and A. Reid that yields surface subgroups of the fundamental group G of the 3-manifold M. By expanding on work of J. Masters and using the structure of the Cannon-Thurston map, we are able to build many quasi-convex surface subgroups of G whose limits sets may be used to separate any pair of distinct points in the boundary of the universal cover of M. As a consequence, G acts geometrically on a finite dimensional CAT(0) cube complex. D. Wise then asks if it is possible that G acts both geometrically and virtually co-specially (in the sense of F. Haglund and D. Wise) on a CAT(0) cube complex. A positive answer would solve the long-standing conjectures that G is large and M has infinite virtual first Betti number. We then explain why finding a virtually embedded cut-and-cross-join surface in a finite cover of M would be enough to solve this problem. Finally, we give some algebraic and then geometric and cohomological sufficient conditions for a given cut-and-cross-join surface to virtually embed.

In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding sought-after group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups G(^m) and G(ⁿ) both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W(²) and W(³) are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. We construct a census of two-generator one-relator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of one-relator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the non-CAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. For every prime p we construct finite p-groups of arbitrary negative deficiency, and thereby complete Kotschick's proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients. We determine examples of sizeable graphs, as required to construct finitely presented non-hyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.