2010/08/09 by Iain R. Aitchison, Aitchison, Iain R.
Mathematics · #2010. Primary: 57M50 #20F67 #52B70 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Mathematics and Applications #Secondary: 51M10 #math.GR #math.GT #msc:2010. #msc:20F67 #msc:51M10 #msc:52B70 #msc:57M50
paper · pdf · doi:10.48550/arxiv.1008.1468
16 pages, 3 figures
arxiv created 2010/08/09 · openalex publication_date 2010/08/09 · arxiv updated 2010/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.