{"paper":{"title":"Mean Dimension, Mean Rank, and von Neumann-L\\\"uck Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.FA","math.GR","math.OA"],"primary_cat":"math.DS","authors_text":"Bingbing Liang, Hanfeng Li","submitted_at":"2013-07-20T22:34:04Z","abstract_excerpt":"We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $\\Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $\\Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-L\\\"uck rank of M all coincide.\n  As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5471","kind":"arxiv","version":3},"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"}