{"paper":{"title":"Kazhdan's Property (T) via Semidefinite Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GR","authors_text":"Andreas Thom, Tim Netzer","submitted_at":"2014-11-10T16:22:50Z","abstract_excerpt":"Following an idea of Ozawa, we give a new proof of Kazhdan's property (T) for ${\\rm SL}(3,\\mathbb Z)$, by showing that $\\Delta^2- \\frac{1}{6} \\Delta$ is a hermitian sum of squares in the group algebra, where $\\Delta$ is the unnormalized Laplace operator with respect to the natural generating set. This corresponds to a spectral gap of $\\frac{1}{72}\\sim 0.014$ for the associated random walk operator.\n  The sum of squares representation was found numerically by a semidefinite programming algorithm, and then turned into an exact symbolic representation, provided in an attached Mathematica file."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2488","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":""},"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"}