On Convex Projective Manifolds and Cusps

2011/09/03 by Daryl Cooper, Cooper, Daryl, Darren Long +3 · 3 citations
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT

paper · pdf · doi:10.48550/arxiv.1109.0585

52 pages, 10 figures, minor corrections and additional references

arxiv created 2012/06/05 · arxiv updated 2012/06/06

Abstract

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space. Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.

Cited by

• Equiaffine immersions and pseudo-Riemannian space forms
• On cusp holonomies in strictly convex projective geometry
• Transverse groups preserving proper domains in flag manifolds

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