{"paper":{"title":"The cycle structure of a Markoff automorphism over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.GR"],"primary_cat":"math.NT","authors_text":"Alois Cerbu, Elijah Gunther, Luke Peilen, Michael Magee","submitted_at":"2016-10-22T17:07:42Z","abstract_excerpt":"We begin an investigation of the action of pseudo-Anosov elements of $\\mathrm{Out}(\\mathbf{F}_{2})$ on the Markoff-type varieties \\[ \\mathbb{X}_{\\kappa}:\\:x^{2}+y^{2}+z^{2}=xyz+2+\\kappa \\] over finite fields $\\mathbb{F}_{p}$ with $p$ prime. We first make a precise conjecture about the permutation group generated by $\\mathrm{Out}(\\mathbf{F}_{2})$ on $\\mathbb{X}_{-2}(\\mathbb{F}_{p})$ that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting `generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov $g\\in\\mathrm{Out}(\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07077","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":""},"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"}