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

We prove that every finite-volume hyperbolic 3-manifold M with p ≥ 1 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K∗ M . 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 Eti × [1,∞), where {Eti } p i=1 is a set of Euclidean cusp tori, with Eti having the canonical shape associated with the ith cusp; (c) Metric singularities are concentrated on the 1-skeleton ofK∗ M with cone angle kπ on any edge of degree k. The CAT(0) 2-complex K∗ M is constructed canonically from Euclidean polygons P e i,j , which reassemble to create {Eti } p i=1; (d) There is a canonical 1-parameter metric deformation, through piecewise-constant-curvature complete metrics, from the hyperbolic metric with limit the piecewise Euclidean one (facilitated by a simple application of Pythagorus’ Theorem); (e) The hyperbolic metric onM can be reconstructed from a finite set of points pi,j on the tori Eti , weighted by real numbers wi,j ∈ (0, 1). Our CAT(0) construction can be considered ‘dual’ to that of Epstein and Penner, but uses much simpler arguments, directly and canonically based on Ford domains. Epstein and Penner’s metrics, parametrized by a choice T of disjoint cusp horotori, gives rise to incomplete piecewise Euclidean metrics with singularities in cusps. To each such choice T , we also construct a complete CAT(0) metric of the above form, with CAT(0) spine KT . This CAT(0) metric structure is already visible via both Weeks’ Snappea program, and its recent manifestation SnapPy by Culler and Dunfield, although its existence has not previously been observed. Our construction also generalizes to finite-volume p-cusped n-manifolds W, to endow each with a complete piecewise-Euclidean CAT(0) metric with non-singular product end structures, whose singularities are concentrated in codimension 2: such W deformation retract to a natural spine, which is CAT(0) as a manifestation of polar duality of ideal hyperbolic polytopes.