Volume gradients and homology in towers of residually-free groups

Abstract

We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups $G_n<G$ of increasing finite index in a fixed finitely generated group $G$, assuming ${\bigcap }_n{G}_n=1$. We focus in particular on finitely presented residually free groups, calculating their ${\ell }_2$ betti numbers, rank gradient and asymptotic deficiency. If $G$ is a limit group and $K$ is any field, then for all $j\ge 1$ the limit of $\dim H_j(G_n,K)/[G,G_n]$ as $n\to \infty$ exists and is zero except for $j=1$, where it equals $-\chi (G)$. We prove a homotopical version of this theorem in which the dimension of $\dim H_j(G_n,K)$ is replaced by the minimal number of $j$-cells in a $K(G_n,1)$; this includes a calculation of the rank gradient and the asymptotic deficiency of $G$. Both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of {\em{slow}} groups. We prove that if a residually free group $G$ is of type ${\mathrm{FP}}_{\mathrm{m}}$ but not of type ${\mathrm{FP}}_{\infty }$, then there exists an exhausting filtration by normal subgroups of finite index $G_n$ so that ${\lim }_n\dim H_j(G_n,K)/[G:G_n]=0forj\le m$. If $G$ is of type ${\mathrm{FP}}_{\infty }$, then the limit exists in all dimensions and we calculate it.

arXiv commentFinal accepted version. To appear in Math Ann