On Convex Projective Manifolds and Cusps

TL;DR

This paper explores convex real projective manifolds, establishing foundational notions like cusps and parabolics, and demonstrating their properties and classifications, including volume and diameter bounds, with implications for hyperbolic geometry.

Contribution

It introduces new concepts such as parabolic, horosphere, and cusp in convex projective manifolds, and proves key results like the Margulis lemma and volume classifications.

Findings

Finite volume cusps are projectively equivalent to hyperbolic cusps.

In dimensions higher than 3, only finitely many topological types exist for bounded volume manifolds.

Diameter bounds relate to the thick part in higher dimensions.

Abstract

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space. Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.