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Every isotopy class $[\\gamma]$ of a closed curve $\\gamma \\in \\pi_{1}(S_{g,n})$ contains a unique closed geodesic on $X$.\n  Let $\\ell_{\\gamma}(X)$ denote the hyperbolic length of the geodesic representative of $\\gamma$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.03342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-13T18:36:08Z","cross_cats_sorted":[],"title_canon_sha256":"900b70fa24c77923b71c15bf39e766f188f65debc2270f4954159777fd5e5321","abstract_canon_sha256":"a9fd616fbf78c060b7eb91ff55078abc6da042766810bcde22e66bf82c563565"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:54.738950Z","signature_b64":"lhKP9JsM8EdvwqfguxTeICmQu0OzjUXxXpdX5KYj5he8GkTicSQUvGHwOXSmRkPaQrqWG1pRdLyGBJLdFK+cDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55342ee98e05cacd4844099596bf394e417957295a4b4de1071fca7c18bb74cf","last_reissued_at":"2026-05-18T01:22:54.738419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:54.738419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting Mapping Class group orbits on hyperbolic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Maryam Mirzakhani","submitted_at":"2016-01-13T18:36:08Z","abstract_excerpt":"Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. 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