Divisible convex sets with properly embedded cones

TL;DR

This paper constructs new examples of properly convex domains divided by Zariski dense relatively hyperbolic groups in higher dimensions, providing insights into convex projective structures with totally geodesic boundaries.

Contribution

It introduces a flexible construction of convex domains with embedded cones, answering Benoist's question and offering a topological criterion for deformations in 3-manifolds.

Findings

Existence of many convex domains divided by hyperbolic groups in all dimensions ≥3

A topological criterion for deformation spaces in 3D convex projective structures

Construction of convex domains with embedded cones capturing hyperbolicity and convexity

Abstract

In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.