Algebraic & Geometric Topology

All flat manifolds are cusps of hyperbolic orbifolds

Darren D Long and Alan W Reid

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We show that all closed flat n–manifolds are diffeomorphic to a cusp cross-section in a finite volume hyperbolic (n+1)–orbifold.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 285-296.

Received: 6 December 2001
Accepted: 10 April 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57R99: None of the above, but in this section

flat manifolds hyperbolic orbifold cusp cross-sections


Long, Darren D; Reid, Alan W. All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol. 2 (2002), no. 1, 285--296. doi:10.2140/agt.2002.2.285.

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