Generalized Cusps in Real Projective Manifolds: Classification

TL;DR

This paper classifies generalized cusps in real projective manifolds by analyzing their geometric structures, group actions, and lattice classifications, revealing conditions for finite Busemann measure and underlying Euclidean structures.

Contribution

It provides a comprehensive classification of generalized cusps using affine group actions and lattice data, linking geometric properties to algebraic structures.

Findings

Cusp classification via affine groups parameterized by Weyl chamber points

Finite Busemann measure linked to unipotent elements in the group

Identification of underlying Euclidean structures independent of Hilbert metric

Abstract

A generalized cusp $C$ is diffeomorphic to $[0,\infty)$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $\Gamma$, in one of a family of affine groups $G(\psi)$, parameterized by a point $\psi$ in the (dual closed) Weyl chamber for $SL(n+1,\mathbb{R})$, and $\Gamma$ determines the cusp up to equivalence. These affine groups correspond to certain fibered geometries, each of which is a bundle over an open simplex with fiber a horoball in hyperbolic space, and the lattices are classified by certain Bieberbach groups plus some auxiliary data. The cusp has finite Busemann measure if and only if $G(\psi)$ contains unipotent elements. There is a natural underlying Euclidean structure on $C$ unrelated to the Hilbert metric.