Generalized Cusps in Real Projective Manifolds: Classification

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.

33 pages, 5 figures This version has been significantly rewritten so that the exposition is improved, details have been added, typos have been corrected, and references added

References in corpus (5)

• Convex projective structures on non-hyperbolic three-manifolds
• Deforming convex projective manifolds
• Convex cocompact actions in real projective geometry
• Convex projective manifolds with a cusp of any non-diagonalizable type
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Cited by in corpus (8)

• Families of Geometries, Real Algebras, and Transitions
• Codimension-$1$ Simplices in Divisible Convex Domains
• Convex projective manifolds with a cusp of any non-diagonalizable type
• Divisible convex sets with properly embedded cones
• Constructing convex projective 3-manifolds with generalized cusps
• The space of properly-convex structures
• On properly convex real-projective manifolds with Generalized Cusp
• The Heisenberg Plane