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This answers in the negative a question asked by Margulis. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.1006","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-09-05T08:40:09Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"5f04d156929bed3dcd568c7115dd72581408c3a8cfdb95e2dccc2720b6cb7575","abstract_canon_sha256":"770312db3004effd8d2226c98746a86c3a61a250246252ab2a5a22f2f2f2b2ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:05.717258Z","signature_b64":"rg90KwT1Ab3UemjuE//b9wW4nzatJPoI4OwpT3KPd3UlpPN4Jtm8sOoe61AjJ3JlDZrvDk+K953zFfMzc3ZyBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c346d223f7c4144720abdef3122e8152c6fe71a41e07bc97f77943e3b907a79d","last_reissued_at":"2026-05-18T02:28:05.716860Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:05.716860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lattices with and lattices without spectral gap","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DS","authors_text":"Alexander Lubotzky, Bachir Bekka","submitted_at":"2009-09-05T08:40:09Z","abstract_excerpt":"The following two results are shown.\n  1) Let $G$ be the $k$-rational points of a simple algebraic group over a local field $k$ and let $H$ be a lattice in $G.$ Then the regular representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with zero mean).\n  2) There exist locally compact simple groups $G$ and lattices $H$ for which $L^2(G/H)$ has no spectral gap. 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