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Euclidean decompositions of noncompact hyperbolic manifolds. (English) Zbl 0611.53036
Suppose that M is an n-dimensional non-compact hyperbolic manifold of finite volume, where \(n\geq 2\). The authors introduce a construction for decomposing M into canonical Euclidean pieces; the method is to work in Minkowski \((n+1)\)-space, represent (horospheres about) the cusps of M as points in the light-cone, and consider the convex hull of all such points P arising from all the cusps. The boundary of this hull is decomposed into affine pieces, each with a canonical Euclidean structure, and this decomposition descends to a corresponding decomposition of M. A crucial fact for this construction is that the set P is discrete, which is somewhat surprising since the corresponding discrete group of Möbius transformations acts ergodically on the light-cone. In particular, when \(n=2\), the decomposition of M is a cell-decomposition.

MSC:
53C20 Global Riemannian geometry, including pinching
53A35 Non-Euclidean differential geometry
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