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Let $G$ be a finite connected graph, $\\mu_1 (G)$ be the linear spectral gap of $G$, and $\\lambda_1 (G,X)$ be the nonlinear spectral gap of $G$ with respect to such a $\\mathrm{CAT}(0)$ space $X$. Then, the result implies that the ratio $\\lambda_1 (G,X) / \\mu_1 (G)$ is bounded from below by a positive constant which is independent of the graph $G$. It follows that any isometric action of a random group of the graph "},"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":"1102.0729","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-03T16:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"af1e0f34a742ed8f63e549516ba44477a747e159f4a44e4a277b55e0e7906f62","abstract_canon_sha256":"dc0edd6e17550369982399aaa400563c58e853f3d1916fddb8160f7028f0aa80"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:12.174671Z","signature_b64":"HJjzxR+BnqZUezs17FYoxeKG+TXP7Ontp+Kg8kxl55357qJB0MpGQ4AEwtjlec1JEf+El5oTuwqTborm+SwVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef2240e13fce118dd81dafdffae268478eb0ea23a55073bb79641df5233d9791","last_reissued_at":"2026-05-18T01:31:12.174027Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:12.174027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fixed point property for a CAT(0) space which admits a proper cocompact group action","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Tetsu Toyoda","submitted_at":"2011-02-03T16:46:54Z","abstract_excerpt":"We prove that if a geodesically complete $\\mathrm{CAT}(0)$ space $X$ admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of $X$ is less than $1$. Let $G$ be a finite connected graph, $\\mu_1 (G)$ be the linear spectral gap of $G$, and $\\lambda_1 (G,X)$ be the nonlinear spectral gap of $G$ with respect to such a $\\mathrm{CAT}(0)$ space $X$. Then, the result implies that the ratio $\\lambda_1 (G,X) / \\mu_1 (G)$ is bounded from below by a positive constant which is independent of the graph $G$. 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