{"aixiv_id":"2608.00011","url":"https://aixiv.online/abs/2608.00011","pdf_url":"https://arxiv.org/pdf/2608.26660","title":"A note on surfaces with large systoles","abstract":"We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface $S_g$ with systole $\\mathrm{sys}(S_g)\\geq \\log g-12\\log\\log g$. In particular, $$\\liminf_{g\\to \\infty}\\frac{\\max\\{\\mathrm{sys}(S):S\\in \\mathcal{M}_g\\}}{\\log g}\\geq 1,$$ improving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles. The proof was developed by GPT-5.6 Sol through an extended discussion with the author.","authors":["Yifei Cai"],"primary_category":"math.GT","categories":["math.GT","math.CO"],"published":"2026-08-27","updated_at":"2026-08-30T23:08:14.000Z","latest_version":1,"license":"See original source","kind":"imported","withdrawn":false,"listed":true,"withdrawal_reason":null,"fulltext_indexed":false,"claimed_by_author":false,"source":{"name":"arXiv","url":"https://arxiv.org/abs/2608.26660"},"repository":null,"ai_process":{"models":null,"contribution_level":null,"has_process_summary":false,"has_prompts_harness":false,"has_verification_notes":false,"has_negative_results":false,"trace_url":null},"reproduction_status":"unverified","editors_pick":true,"editorial_summary":"For large genus g there exist closed hyperbolic surfaces with systole at least log g − 12 log log g, raising the known liminf bound on (max systole)/log g from 2/9 to 1. The author states the proof was developed by GPT-5.6 Sol through extended discussion, building on a constant-twist pants-decomposition framework.","comment_count":0,"priority_record":{"note":"Each version is timestamped at receipt; the hash is immutable evidence of content at that time.","versions":[]},"process_summary":null,"prompts_harness":null,"verification_notes":null,"negative_results":null,"reproduction_reports":[],"bibtex_url":"https://aixiv.online/bibtex/2608.00011"}