All flat manifolds are cusps of hyperbolic orbifolds

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R , C , or H , it is well known that cusp cross-sections of finite volume X –hyperbolic (n + 1)–orbifolds are flat n–orbifolds or almost flat orbifolds modelled on the (2n + 1)–dimensional Heisenberg group N2n+1 or the (4n + 3)–dimensional quaternionic Heisenberg group N4n+3(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section o...

All at Manifolds Are Cusps of Hyperbolic Orbifolds

All nil 3-manifolds are cusps of complex hyperbolic 2-orbifolds

In this paper, we prove that every closed nil 3-manifold is diffeomorphic to a cusp cross-section of a finite volume complex hyperbolic 2-orbifold.

Arithmetic cusp shapes are dense

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n–manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n+1)–orbifold is dense in the space of similarity classes of flat metrics on M. The set used for density is precisely the set of those classes which arise i...

Cusps of Minimal Non-compact Arithmetic Hyperbolic 3-orbifolds

In this paper we count the number of cusps of minimal non-compact finite volume arithmetic hyperbolic 3-orbifolds. We show that for each N , the orbifolds of this kind which have exactly N cusps lie in a finite set of commensurability classes, but either an empty or an infinite number of isometry classes.