{"paper":{"title":"Spherical growth of reciprocal classes in the Hecke Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Debattam Das, Krishnendu Gongopadhyay","submitted_at":"2024-11-01T17:05:22Z","abstract_excerpt":"Let $\\Gamma_p$ denote the Hecke group where $p=2r$, $r>0$. Let $\\mathcal{N}_l$ denote the set of conjugacy classes of reciprocal elements of word length $l$ in $\\Gamma_p$. We prove that for $l \\to \\infty$,\n  $$|\\mathcal{N}_l| = \\mathcal{O}\\left(\\left\\lfloor \\tfrac{l+1}{2} \\right\\rfloor^{s-1} \\rho^{\\left\\lfloor \\tfrac{l+1}{2} \\right\\rfloor} \\right),\n  $$\n  where $\\mathcal O$ is the `big O', $\\rho \\in [\\sqrt{2}, 2]$ is the unique positive real root of\n  $$\n  p(x) = x^{r+1} - 2\\sum_{j=1}^{r-1} x^{r-j} - 1,\n  $$\n  and $s$ is the maximal multiplicity among the roots of $p(x)$.\n  Our method relies o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.00739","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.00739/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}