{"paper":{"title":"Edge zeta function and closed cycles in the standard non-uniform complex from $\\operatorname{PGL}_3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.DS","math.GT","math.NT","math.RT"],"primary_cat":"math.GR","authors_text":"Sanghoon Kwon, Soonki Hong","submitted_at":"2024-11-23T08:16:58Z","abstract_excerpt":"In this paper, we define the edge zeta function of weighted complex. We also present the formula for the edge zeta function of the standard non-uniform complex $\\operatorname{PGL}(3,\\mathbb{F}_q[t])\\backslash\\operatorname{PGL}(3,\\mathbb{F}_q(\\!(t^{-1})\\!))/\\operatorname{PGL}(3,\\mathbb{F}_q[\\![t^{-1}]\\!])$, arising from the group $\\operatorname{PGL}_3$, as a rational function. Applying trunction in a specific direction is one of the main ingredient. As a result, we obtain the exact formula for the number of closed cycles coming from geodesics in the building."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15489","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15489/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"}