UC Berkeley

This thesis consists of two projects related to groups acting on metric spaces of non-positive curvature.

In the first project, we show that relatively hyperbolic groups acting properly and cocompactly on $\mathrm{CAT}(0)$ cube complexes are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for cubulated hyperbolic groups (which led to the proof of the Virtual Haken Conjecture), and applies to a wide range of peripheral subgroups. In particular, we deduce virtual specialness for properly and cocompactly cubulated groups that are hyperbolic relative to virtually abelian groups. As another consequence, by using a theorem of Martin and Steenbock we obtain virtual specialness for $C'(1/6)$-small cancellation quotients of free products of virtually special groups. For the proof of our main result, we prove a relative version of Wise's quasiconvex hierarchy theorem.

In our second project, we study the metric and topological properties of the space $\mathscr{D}_{\Gamma}$ of metric structures on the non-elementary hyperbolic group $\Gamma$, which parametrizes geometric actions of $\Gamma$ on Gromov hyperbolic spaces. This space contains the Teichm\"uller space when $\Gamma$ is a surface group and the Culler-Vogtmann outer space when $\Gamma$ is a free group. Equipped with a natural metric reminiscent of Thurston's metric on Teichm\"uller space, we prove that $\mathscr{D}_{\Gamma}$ is unbounded, contractible and separable, and that $\mathrm{Out}(\Gamma)$ acts metrically properly by isometries on it. If we restrict to the subspace $\mathscr{D}^{\delta}_{\Gamma}$ of the points represented by actions on $\delta$-hyperbolic spaces with exponential growth rate 1, we prove that it is either empty or proper, and that the Bowen-Margulis map from $\mathscr{D}^{\delta}_{\Gamma}$ into the space $\mathbb{P}\mathcal{C}urr(\Gamma)$ of projective geodesic currents on $\Gamma$ is continuous. By finding an $\mathrm{Out}(\Gamma)$-invariant geodesic bicombing for $\mathscr{D}_{\Gamma}$ we also construct a boundary for this space, which parametrizes improper actions of $\Gamma$ on hyperbolic spaces. As a corollary of this construction, we deduce continuous extension of translation length functions to the space of geodesic currents, which we use to disprove a conjecture of Bonahon about small actions of hyperbolic groups on $\mathbb{R}$-trees.