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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn036, 38 pages, doi:10.1093/imrn/rnn036 published on May 6, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Covering Spaces of Arithmetic 3-Orbifolds

Marc Lackenby1, Darren D. Long2 and Alan W. Reid3

1 Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK
2 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
3 Department of Mathematics, University of Texas, Austin, TX 78712, USA

Correspondence: Correspondence to be sent to: areid{at}math.utexas.edu

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 A4, S4 or A5. 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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