{"paper":{"title":"On a Class of Type II$_1$ Factors with Betti Numbers Invariants","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.OA","authors_text":"Sorin Popa","submitted_at":"2002-09-11T16:17:55Z","abstract_excerpt":"We prove that a type II$_1$ factor $M$ can have at most one Cartan subalgebra $A$ satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class $\\Cal H \\Cal T$ of factors $M$ having such Cartan subalgebras $A \\subset M$, the Betti numbers of the standard equivalence relation associated with $A \\subset M$ ([G2]), are in fact isomorphism invariants for the factors $M$, $\\beta^{^{HT}}_n(M), n\\geq 0$. The class $\\Cal H\\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $\\beta^{^{HT}}_n(M^t)= \\be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0209130","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0209130/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}