{"paper":{"title":"The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\\mathbb{Z}[B\\_4(\\mathbb{S}^2)]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.KT","authors_text":"CNRS), Daniel Juan-Pineda (CCM), John Guaschi (LMNO, NU, Silvia Mill\\'an-L\\'opez, UNICAEN","submitted_at":"2012-09-21T11:46:07Z","abstract_excerpt":"We study $K$-theoretical aspects of the braid groups $B\\_n(\\mathbb{S}^{2})$ on $n$ strings of the $2$-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture~\\cite{JM}. In light of this, in order to determine the algebraic $K$-theory of the group ring $\\mathbb{Z}[B\\_n(\\mathbb{S}^{2})]$, one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gon{\\c c}alves and the first author. We calculate the Whitehead and $K\\_{-1}$-groups of the group rings of the finite subgroups (dicyclic and binary po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4791","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"}