The Moduli Space of Marked Generalized Cusps in Real Projective Manifolds

TL;DR

This paper studies the moduli space of generalized cusps in real projective manifolds, classifying their structures and describing the space's topology and geometric properties, including fiber structures related to Euclidean and cubic differential data.

Contribution

It provides a detailed classification of marked generalized cusps and describes the topology and geometric structure of their moduli space, including fiber descriptions involving cubic differentials.

Findings

The moduli space is homeomorphic to a subspace of conjugacy classes of representations.

It has a description as a generalized trace-variety and involves weight data similar to semi-simple Lie groups.

For 3D orientable cusps, the fiber is a cone on a solid torus.

Abstract

In this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to $M \times [0, \infty)$ where $M$ is a closed Euclidean manifold. These are classified in [2]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of $\pi_1(M)$. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on $M$, and the fiber is a closed cone in the space of cubic differentials. For 3-dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.