{"paper":{"title":"Surface Words are Determined by Word Measures on Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Doron Puder, Michael Magee","submitted_at":"2019-02-13T12:22:38Z","abstract_excerpt":"Every word $w$ in a free group naturally induces a probability measure on every compact group $G$. For example, if $w=\\left[x,y\\right]$ is the commutator word, a random element sampled by the $w$-measure is given by the commutator $\\left[g,h\\right]$ of two independent, Haar-random elements of $G$. Back in 1896, Frobenius showed that if $G$ is a finite group and $\\psi$ an irreducible character, then the expected value of $\\psi\\left(\\left[g,h\\right]\\right)$ is $\\frac{1}{\\psi\\left(e\\right)}$. This is true for any compact group, and completely determines the $\\left[x,y\\right]$-measure on these gro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.04873","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/1902.04873/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"}