Large groups, Property (tau) and the homology growth of subgroups

Details Large groups, Property (tau) and the homology growth of subgroups Large groups, Property (tau) and the homology growth of subgroups

2005-09-02

arxiv.org/abs/math/0509036v3

Mathematics

Group Theory

35 pages, 2 figures; minor corrections and improvements from version 2

Scientific paper

We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of d_p(G_i)/[G:G_i] is positive. We show that if this holds and each G_i is normal in its predecessor and has index a power of p, then one of the following possibilities must be true: G is large (that is, some finite index subgroup admits a surjective homomorphism onto a non-abelian free group) or G has Property (tau) with respect to {G_i}. The arguments are based on the geometry and topology of finite 2-complexes. This has several consequences. It implies that if the pro-p completion of a finitely presented group G has exponential subgroup growth, then G has Property (tau) with respect to some nested sequence of finite index subgroups. It also has applications to low-dimensional topology. We use it to prove that a group-theoretic conjecture of Lubotzky-Zelmanov would imply the following: any lattice in PSL(2,C) with torsion is large. We also relate linear growth of mod p homology to the existence of certain important error-correcting codes: those that are `asymptotically good', which means that they have large rate and large Hamming distance.

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