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Some aspects of geometric actions of hyperbolic and relatively hyperbolic groups
 Oregon Reyes, Eduardo
 Advisor(s): Agol, Ian
Abstract
This thesis consists of two projects related to groups acting on metric spaces of nonpositive 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 nonelementary 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 CullerVogtmann 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 BowenMargulis 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.
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