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Proceedings of the London Mathematical Society Advance Access originally published online on July 10, 2008
Proceedings of the London Mathematical Society 2009 98(2):271-297; doi:10.1112/plms/pdn032
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© 2008 London Mathematical Society

New lower bounds on subgroup growth and homology growth

Marc Lackenby

Mathematical Institute
University of Oxford
24-29 St Giles’
Oxford
OX1 3LB
United Kingdom

Received 14 March 2008. Revision received 1 May 2008.

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subgroup growth of any finitely generated group. This is achieved by free non-abelian groups and is slightly faster than exponential. As a consequence, we obtain good estimates on the number of covering spaces of a hyperbolic 3-manifold with given covering degree. We also obtain slightly weaker information on the number of covering spaces of closed 4-manifolds with non-positive Euler characteristic. The results on subgroup growth follow from a new theorem, which places lower bounds on the rank of the first homology (with mod p coefficients) of certain subgroups of a group. This is proved using a topological argument.

2000 Mathematics Subject Classification 20E07, 57M10, 57N10, 57N13.

The research is supported by an EPSRC Advanced Fellowship.


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