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All flat manifolds are cusps of hyperbolic orbifolds. (English) Zbl 0998.57038
The authors prove that for every \(n \geq 2\), the diffeomorphism class of every flat \(n\)-manifold has a representative \(W\) which arises as some cusp cross-section of a finite volume cusped hyperbolic \((n+1)\)-orbifold. This is a partial answer to a question posed by Farrell and Zdravkovska which asked if the diffeomorphism class of every flat \(n\)-manifold has a representative \(W\) which arises as the cusp cross-section of a finite volume \(1\)-cusped hyperbolic \(n\)-manifold. However, the authors have been unable to pass to manifolds from orbifolds, or to guarantee that their examples are \(1\)-cusped. The main ingredient of their proof is arithmetic, the main technical result being that the fundamental group of a flat \(n\)-manifold embeds as a subgroup of \(O_{0}(q_{n+2};{\mathbb Z})\), for some quadratic form \(q_{n+2}\) defined over \({\mathbb Q}\) of signature \((n+1,1)\). This together with some separability arguments allows the authors to prove the main result.

MSC:
57M50 General geometric structures on low-dimensional manifolds
57R99 Differential topology
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