New lower bounds on subgroup growth and homology growth

Details New lower bounds on subgroup growth and homology growth New lower bounds on subgroup growth and homology growth

2005-12-13

arxiv.org/abs/math/0512261v3

Mathematics

Group Theory

39 pages, 2 figures; v3 has minor changes from v2, incorporating referee's comments; v2 has minor changes from v1; to appear i

Scientific paper

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.

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