{"paper":{"title":"Hausdorff dimension and Kleinian groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Christopher J. Bishop, Peter Jones","submitted_at":"1994-03-22T00:00:00Z","abstract_excerpt":"Let G be a non-elementary, finitely generated Kleinian group, Lambda(G) its limit set and Omega(G) = S \\ Lambda(G) (S = the sphere) its set of discontinuity. Let delta(G) be the critical exponent for the Poincar\\'e series and let Lambda_c be the conical limit set of G. Suppose Omega_0 is a simply connected component of Omega(G). We prove that\n  (1) delta(G) = dim(Lambda_c).\n  (2) A simply connected component Omega is either a disk or dim(Omega)>1$.\n  (3) Lambda(G) is either totally disconnected, a circle or has dimension > 1,\n  (4) G is geometrically infinite iff dim(Lambda)=2.\n  (5) If G_n \\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9403222","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"}