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New lower bounds on subgroup growth and homology growth
Author(s) -
Lackenby Marc
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn032
Subject(s) - mathematics , homology (biology) , combinatorics , pure mathematics , biology , genetics , gene
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.