Generalized Cusps in Real Projective Manifolds: Classification

A generalized cusp $C$ is diffeomorphic to $[0,\mathrm{\infty })$ times a closed Euclidean manifold. Geometrically $C$ is the quotient of a properly convex domain by a lattice, $\mathrm{\Gamma }$, 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 $\mathrm{\Gamma }$ 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.