A generalization of the Epstein-Penner construction to projective manifolds

TL;DR

This paper generalizes Epstein-Penner's hyperbolic cell decomposition to strictly convex projective manifolds, showing small holonomy deformations yield nearby structures with radial ends under certain conditions.

Contribution

It extends the canonical cell decomposition to the convex projective setting and establishes conditions for deformations to produce structures with radial ends.

Findings

Extended Epstein-Penner decomposition to convex projective manifolds

Proved small holonomy deformations lead to nearby structures with radial ends

Identified fixed point conditions for cusp holonomies

Abstract

We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with radial ends, provided the holonomy of each cusp has a fixed point.