Geometric structures on negatively curved groups and their subgroups

In this thesis, we investigate two explicit families of geometric structures that occur on hy- perbolic groups. After recalling some introductory material, we begin by giving an overview of the theory of special cube complexes, with a particular focus on properties of subgroups of hyperbolic special groups. We then describe an explicit algorithm, based on Stallings’ notion of folding for graphs, to construct a local isometry between cube complexes that represents the inclusion of a subgroup H ⊂ G , and show that this terminates if and only if the subgroup is quasiconvex. This provides a potential method by which quasiconvexity for various sub- groups could be verified. In the second part of the thesis, we investigate another family of geometric structures: negatively curved simplicial complexes. We show that groups satisfying a “uniform” C′(1/6) small cancellation condition have such a structure, and then move on to prove a gluing theo- rem (with cyclic edge groups) for these complexes. Using this theorem, we extend the family of groups known to be CAT(−1) to include hyperbolic limit groups, hyperbolic graphs of free groups with cyclic edge groups, and more generally hyperbolic groups whose JSJ components are 2-dimensionally CAT(−1)