Generalized Cusps in Real Projective Manifolds: Classification,
Sam Ballas, Daryl Cooper, Arielle Leitner
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, 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.