Volume gradients and homology in towers of residually-free groups

TL;DR

This paper investigates the asymptotic behavior of homology groups and cellular volumes in towers of residually free groups, providing new calculations of invariants like Betti numbers, rank gradient, and deficiency, with both homological and homotopical perspectives.

Contribution

It introduces new results on the asymptotic growth of homology and homotopy invariants in residually free groups, including limits of homology dimensions and a homotopical version of these theorems.

Findings

Limit of homology dimension ratios exists and is zero except for j=1.

Calculated Betti numbers, rank gradient, and deficiency for residually free groups.

Established results as special cases of fundamental groups of graphs of slow 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 $\rm{FP}_m$ but not of type $\rm{FP}_{\infty}$, then there exists an exhausting filtration by normal subgroups of finite index $G_n$ so that $\lim_n \dim H_j (G_n, K) / [G : G_n] = 0 \hbox{for} j \leq m$. If $G$ is of type $\rm{FP}_{\infty}$, then the limit exists in all dimensions and we calculate it.