{"paper":{"title":"Property (T) for groups graded by root systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.RT"],"primary_cat":"math.GR","authors_text":"Andrei Jaikin-Zapirain, Martin Kassabov, Mikhail Ershov","submitted_at":"2011-01-31T23:07:29Z","abstract_excerpt":"We introduce and study the class of groups graded by root systems. We prove that if {\\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\\Phi}, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem we prove that for any reduced irreducible classical root system {\\Phi} of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_{\\Phi}(R) and the elementary Chevalley group E_{\\Phi}(R) have property (T). We also show that ther"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0031","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":""},"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"}