{"paper":{"title":"Soficity, short cycles and the Higman group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Harald A. Helfgott, Kate Juschenko","submitted_at":"2015-12-07T17:36:03Z","abstract_excerpt":"This is a paper with two aims. First, we show that the map from $\\mathbb{Z}/p\\mathbb{Z}$ to itself defined by exponentiation $x\\to m^x$ has few $3$-cycles -- that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds.\n  Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\\mathbb{Z}/p\\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02135","kind":"arxiv","version":4},"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"}