{"paper":{"title":"Building hyperbolic metrics suited to closed curves and applications to lifting simply","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jenya Sapir, Jonah Gaster, Priyam Patel, Tarik Aougab","submitted_at":"2016-03-21T01:26:42Z","abstract_excerpt":"Let $\\gamma$ be an essential closed curve with at most $k$ self-intersections on a surface $\\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $\\rho$ for which $\\gamma$ has length at most $M \\cdot \\sqrt{k}$, where $M$ is a constant depending only on the topology of $\\mathcal{S}$. Moreover, the injectivity radius of $\\rho$ is at least $1/(2\\sqrt{k})$. This yields linear upper bounds in terms of self-intersection number on the minimum degree of a cover to which $\\gamma$ lifts as a simple closed curve (i.e. lifts simply). We also show that if $\\gamma$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06303","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"}