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

TL;DR

This paper demonstrates that finite-volume hyperbolic 3-manifolds with cusps can be equipped with a canonical CAT(0) metric, revealing new geometric structures and deformations related to their canonical decompositions.

Contribution

It introduces a canonical CAT(0) piecewise Euclidean metric on hyperbolic 3-manifolds with cusps, extending the Epstein-Penner decomposition and providing a deformation from hyperbolic to CAT(0) geometry.

Findings

Universal cover is a union of Euclidean half-spaces

Cusps are non-singular Euclidean products with half-lines

Singularities are concentrated on the 1-skeleton with cone angles multiple of pi

Abstract

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.