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

Abstract

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.