{"paper":{"title":"Geometric invariants of locally compact groups: the homotopical perspective","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Elisa Hartmann, Jos\\'e Pedro Quintanilha, Kai-Uwe Bux","submitted_at":"2024-10-25T12:02:24Z","abstract_excerpt":"We extend the classical theory of homotopical $\\Sigma$-sets $\\Sigma^n$ developed by Bieri, Neumann, Renz and Strebel for abstract groups, to $\\Sigma$-sets $\\Sigma_{\\mathrm{top}}^n$ for locally compact Hausdorff groups. Given such a group $G$, our $\\Sigma_{\\mathrm{top}}^n(G)$ are sets of continuous homomorphisms $G \\to \\mathbb{R}$ (\"characters\"). They match the classical $\\Sigma$-sets $\\Sigma^n(G)$ if $G$ is discrete, and refine the homotopical compactness properties $\\mathrm C_n$ of Abels and Tiemeyer. Moreover, our theory recovers the definition of $\\Sigma_{\\mathrm{top}}^1$ and $\\Sigma_{\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.19501","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.19501/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"}