Cusped hyperbolic 3-manifolds: canonically CAT(0) with CAT(0) spines

Details Cusped hyperbolic 3-manifolds: canonically CAT(0) with CAT(0) spines Cusped hyperbolic 3-manifolds: canonically CAT(0) with CAT(0) spines

2010-08-09

arxiv.org/abs/1008.1468v1

Mathematics

Geometric Topology

16 pages, 3 figures

Scientific paper

We prove that every finite-volume hyperbolic 3-manifold M with p > 0 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K. Moreover, (a) the universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identifying Euclidean polygons in their bounding planes by pairwise isometry (b)each cusp of M in the CAT(0) metric is a non-singular metric product of a (Euclidean) cusp torus and a half-line (c) all metric singularities are concentrated on the 1-skeleton of K, with cone angles a multiple of pi (d) there is a canonical deformation of the hyperbolic metric with limit the CAT(0) piecewise Euclidean metric. The proof uses Ford domains; the construction is essentially the polar-dual of the Epstein-Penner canonical decomposition, and generalizes to higher dimension.

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