In this dissertation, we study the Hilbert geometry of properly convex domains from the perspective of non-positive curvature and geometric group theory. First, we introduce a notion of rank one properly convex domains and prove that rank one groups are either acylindrically hyperbolic or contain a finite index cyclic subgroup. This is in the spirit of rank one non-positively curved Riemannian manifolds. Second, we develop the notion of “properly convex domains with strongly isolated simplices” which is a finer notion than rank one. We prove that this notion completely characterizes the relative hyperbolicity of convex co-compact groups. This answers a question of Danciger-Gu´eritaud-Kassel and provides a possible research direction for generalizing Anosov representation beyond Gromov hyperbolic groups. Lastly, we establish a convex projective analogue of the well-known Flat Torus Theorem from CAT(0) geometry.

Mitul's advisor is Ralf Spatzier.

Zoom: https://umich.zoom.us/j/97424051440
Passcode: defense Speaker(s): Mitul Islam (UM)