{"paper":{"title":"On the hyperbolic distance of $n$-times punctured spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Matti Vuorinen, Tanran Zhang, Toshiyuki Sugawa","submitted_at":"2017-07-18T17:51:55Z","abstract_excerpt":"The length of the shortest closed geodesic in a hyperbolic surface $X$ is called the systole of $X.$ When $X$ is an $n$-times punctured sphere $\\hat{ \\mathbb{C}} \\setminus A$ where $A \\subset \\hat{\\mathbb{C}}$ is a finite set of cardinality $n\\ge4,$ we define a quantity $Q(A)$ in terms of cross ratios of quadruples in $A$ so that $Q(A)$ is quantitatively comparable with the systole of $X.$ We next propose a method to construct a distance function $d_X$ on a punctured sphere $X$ which is Lipschitz equivalent to the hyperbolic distance $h_X$ on $X.$ In particular, when the construction is based "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.05773","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"}