{"paper":{"title":"Counting curves, and the stable length of currents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.DS","math.GR"],"primary_cat":"math.GT","authors_text":"Hugo Parlier, Juan Souto, Viveka Erlandsson","submitted_at":"2016-12-18T19:54:06Z","abstract_excerpt":"Let $\\gamma_0$ be a curve on a surface $\\Sigma$ of genus $g$ and with $r$ boundary components and let $\\pi_1(\\Sigma)\\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $\\gamma$ of type $\\gamma_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $\\pi_1(\\Sigma)$ the limit $$\\lim_{L\\to\\infty}\\frac 1{L^{6g-6+2r}}\\{\\gamma\\text{ of type }\\gamma_0\\text{ with }S\\text{-translation length}\\le L\\}$$ exists and is positive. The main new technical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05980","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"}