{"paper":{"title":"Topology and convexity in the space of actions modulo weak equivalence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.LO"],"primary_cat":"math.DS","authors_text":"Peter Burton","submitted_at":"2015-01-16T19:09:06Z","abstract_excerpt":"We analyse the structure of the quotient $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We show that the convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is compatible with the topology, and as a consequence deduce that $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ for amenable $\\Gamma$ by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04079","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"}