{"paper":{"title":"Thurston norm, polytopes and splitting complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Andrei Jaikin-Zapirain, Monika Kudlinska, Pablo S\\'anchez-Peralta","submitted_at":"2026-06-30T14:57:29Z","abstract_excerpt":"We show that if $G$ is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first $L^{2}$-Betti number, then the map that assigns to each surjective integral character the first $L^2$-Betti number of the kernel extends to a seminorm on the first cohomology group of $G$ with real coefficients. We call this seminorm the Thurston norm. Moreover, we show that this norm is induced by a polytope in the first homology group with real coefficients. We also generalize this result to higher $L^{2}$-Betti numbers of the kernels, thereby confirming a conjecture of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31774","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.31774/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"}