Controlling manifold covers of orbifolds

Abstract

Abstract. In this article we prove a generalization of Selberg's lemma on the existence of torsion free, finite index subgroups of arithmetic groups. Some of the geometric applications are the resolution a conjecture of Nimershiem and answers to questions of Long-Reid and the author.

Virtually spinning hyperbolic manifolds

“…Let Y be one of these non-spinnable orientable flat 4-manifolds. By [12] and the improvement in [14], every flat 4-manifold occurs as some cusp cross-section of a possibly (indeed, likely) multi-cusped arithmetic hyperbolic 5-manifold. Hence Y can be arranged as a cusp cross-section of an arithmetic hyperbolic 5-manifold X. X cannot be spin, since it is well known that a spin structure induces a spin structure on a boundary component (see [11,Chapter II,Proposition 2.15]).…”

Embedding arithmetic hyperbolic manifolds

“…7 Completing the proof of Theorem 1.1 Given Proposition 5.1 and Corollary 4.2, the following result will complete the proof of Theorem 1.1. The proof of Proposition 7.1 is given in a much more general context in [24], however, for completeness we give a proof in §7.1 adapted to the case at hand. Proposition 7.1.…”

Hyperbolic orbifolds of small volume

“…Long and Reid showed that any closed flat (n − 1)-manifold N is diffeomorphic to a cusp cross-section of a finite volume hyperbolic n-orbifold M [37]. It was later proved by McReynolds that the same can be achieved with M being a manifold [39]. In both constructions the resulting n-orbifold or manifold can be chosen to be arithmetic.…”