{"paper":{"title":"A presentation of symplectic Steinberg modules and cohomology of $\\operatorname{Sp}_{2n}(\\mathbb{Z})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.GR","math.GT","math.NT"],"primary_cat":"math.AT","authors_text":"Benjamin Br\\\"uck, Peter Patzt, Robin J. Sroka","submitted_at":"2023-06-05T18:41:18Z","abstract_excerpt":"Borel-Serre proved that the integral symplectic group $\\operatorname{Sp}_{2n}(\\mathbb{Z})$ is a virtual duality group of dimension $n^2$ and that the symplectic Steinberg module $\\operatorname{St}^\\omega_n(\\mathbb{Q})$ is its dualising module. This module is the top-dimensional homology of the Tits building associated to $\\operatorname{Sp}_{2n}(\\mathbb{Q})$. We find a presentation of this Steinberg module and use it to show that the codimension-1 rational cohomology of $\\operatorname{Sp}_{2n}(\\mathbb{Z})$ vanishes for $n \\geq 2$, $H^{n^2 -1}(\\operatorname{Sp}_{2n}(\\mathbb{Z});\\mathbb{Q}) \\cong"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.03180","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/2306.03180/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"}