Relatively Dominated Representations

Abstract

Convex cocompact subgroups of rank-one semisimple Lie groups such as PSL(2,R) form a structurally stable class of quasi-isometrically embedded discrete subgroups which are naturally associated to negatively-curved geometric structures. Anosov representations give a higher-rank analogue of convex cocompactness which shares many of its good geometric and dynamical properties, and have become important objects of study in higher Teichmueller theory.

In this thesis we introduce relatively dominated representations as a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness---a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity.

We prove that groups admitting relatively dominated representations must be relatively hyperbolic, that these representations induce limit maps with good properties, provide examples, and draw connections to work of Kapovich--Leeb which also introduces higher-rank analogues of geometric finiteness.