{"paper":{"title":"Vanishing of $L^{2}$-Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.GT"],"primary_cat":"math.AT","authors_text":"Feng Ji, Shengkui Ye","submitted_at":"2017-03-01T02:34:20Z","abstract_excerpt":"Let $R$ be an infinite commutative ring with identity and $n\\geq 2$ be an integer. We prove that for each integer $i=0,1,\\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\\ $when $G=\\mathrm{GL}_{n}(R)$ the general linear group, $\\mathrm{SL}_{n}(R)$ the special linear group, $% E_{n}(R)$ the group generated by elementary matrices. When $R$ is an infinite principal ideal domain, similar results are obtained for $\\mathrm{Sp}_{2n}(R)$ the symplectic group, $\\mathrm{ESp}_{2n}(R)$ the elementary symplectic group, $\\mathrm{O}(n,n)(R)$ the split orthogonal group or $\\mathrm{EO}(n,n)(R)$ the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00107","kind":"arxiv","version":1},"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"}