Constructing convex projective 3-manifolds with generalized cusps

TL;DR

This paper demonstrates how certain hyperbolic 3-manifolds can be deformed into convex projective structures with generalized cusps, including the first example of a 1-cusped manifold with a type 2 cusp.

Contribution

It introduces methods to produce convex projective structures with specific cusp types on hyperbolic 3-manifolds, including the first known example with a type 2 cusp.

Findings

Existence of convex projective structures with generalized cusps on hyperbolic 3-manifolds.

Construction of the first 1-cusped hyperbolic 3-manifold with a type 2 cusp.

Development of techniques to control cusp types in convex projective deformations.

Abstract

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling which types of cusp occur. Using these methods we produce the first known example of a 1-cusped hyperbolic 3-manifold that admits a convex projective structure with a type 2 cusp. We also use these techniques to produce new 1-cusped manifolds that admit a convex projective structure with a type 1 cusp.