Covering spaces of arithmetic 3-orbifolds

Marc Lackenby; Darren D. Long; Alan W. Reid

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arXiv · math/0601677

Covering spaces of arithmetic 3-orbifolds

Marc Lackenby·Darren D. Long·Alan W. Reid

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Abstract

Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has infinite virtual first Betti number and G has super-exponential subgroup growth. Our first result, which forms the basis for the entire paper, is that G is commensurable with a lattice containing the Klein group of order 4. Our second result is that if G has a finite index subgroup with first Betti number at least 4, then G is large. This is known to hold for many arithmetic lattices. In particular, we show that it is always the case when G contains A_4, S_4 or A_5. Our third main result is that the Lubotzky-Sarnak conjecture and the geometrisation conjecture together imply that any arithmetic Kleinian group G is large. We also give a new 'elementary' proof that arithmetic Kleinian groups do not have the congruence subgroup property, which avoids the use of the Golod-Shafarevich inequality.

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BibTeX RIS

Marc Lackenby, Darren D. Long, Alan W. Reid. 2008-04-09. Covering spaces of arithmetic 3-orbifolds. https://arxiv.org/abs/math/0601677

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