Large groups, property (τ) and the homology growth of subgroups

Abstract

We investigate the homology of finite index subgroups Gi of a given finitely presented group G. We fix a prime p, denote the field of order p by p and define dp(Gi) to be the dimension of H1(Gi; p). We will be interested in the situation where dp(Gi) grows fast as a function of the index [G : Gi]. Specifically, we say that a collection of finite index subgroups {Gi} has linear growth of mod p homology if infidp(Gi)/[G : Gi] is positive. This is a natural and interesting condition that arises in several different contexts. For example, the main theorem of [9] states that when G is a lattice in PSL(2, ℂ) with non-trivial torsion (equivalently, G is the fundamental group of a finite-volume hyperbolic 3-orbifold with non-empty singular locus), then G has such a sequence of subgroups. Another major class of groups G having such a collection of subgroups are those that are large. By definition, this means that G has a finite index subgroup that admits a surjective homomorphism onto a free non-abelian group. Large groups have many nice properties, for example super-exponential subgroup growth and infinite virtual first Betti number. One might wonder whether largeness is equivalent to the existence of some nested sequence of finite index subgroups {Gi} with linear growth of mod p homology for some prime p. If so, this would establish that lattices in PSL(2, ℂ) with non-trivial torsion are large, which would be a major breakthrough in low-dimensional topology.