{"paper":{"title":"Word-Induced Measures on Compact Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.PR","math.RT"],"primary_cat":"math.GR","authors_text":"Gene S. Kopp, John D. Wiltshire-Gordon","submitted_at":"2011-02-21T21:38:56Z","abstract_excerpt":"Consider a group word w in n letters. For a compact group G, w induces a map G^n \\rightarrow G$ and thus a pushforward measure {\\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that {\\mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsen's classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {\\mu}_w that were derived by Witten in the context of quantum gauge theory [Wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4353","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"}