Examples of extended geometrically finite representations

TL;DR

This paper explores extended geometrically finite (EGF) representations, demonstrating their occurrence in convex projective manifolds and their stability under small deformations, advancing understanding of geometric structures in hyperbolic groups.

Contribution

It proves that holonomy representations of certain convex cocompact manifolds are EGF and characterizes EGF representations as arising from convex projective structures and compositions with symmetric representations.

Findings

Holonomy of convex cocompact manifolds with relatively hyperbolic groups are EGF.

EGF representations correspond to holonomies of convex projective manifolds with generalized cusps.

Small deformations of certain representations remain EGF.

Abstract

This is the second of a pair of papers on extended geometrically finite (EGF) representations, which were originally posted as a single article under the title "An extended definition of Anosov representation for relatively hyperbolic groups." In this paper, we prove that the holonomy representation of a projectively convex cocompact manifold with relatively hyperbolic fundamental group is always an EGF representation. We also prove that EGF representations arise as holonomy representations of convex projective manifolds with generalized cusps and as compositions of projectively convex cocompact representations with symmetric representations of SL(d, R). We additionally show that any small deformation of a representation of the latter form is still EGF.