{"paper":{"title":"Volume gradients and homology in towers of residually-free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.KT"],"primary_cat":"math.GR","authors_text":"Dessislava H. Kochloukova, Martin R Bridson","submitted_at":"2013-09-07T15:52:23Z","abstract_excerpt":"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.\n  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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1877","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}