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• W4298026907 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 $jge 1$ the limit of $dim H_j(G_n,K)/[G,G_n]$ as $ntoinfty$ 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." @default.
• W4298026907 created "2022-10-01" @default.
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• W4298026907 date "2013-09-07" @default.
• W4298026907 modified "2023-09-27" @default.
• W4298026907 title "Volume gradients and homology in towers of residually-free groups" @default.
• W4298026907 doi "https://doi.org/10.48550/arxiv.1309.1877" @default.
• W4298026907 hasPublicationYear "2013" @default.
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