{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:54REBYJ7ZYIY3WA5V7P7VYTII6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"dc0edd6e17550369982399aaa400563c58e853f3d1916fddb8160f7028f0aa80","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-03T16:46:54Z","title_canon_sha256":"af1e0f34a742ed8f63e549516ba44477a747e159f4a44e4a277b55e0e7906f62"},"schema_version":"1.0","source":{"id":"1102.0729","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0729","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0729v4","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0729","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"pith_short_12","alias_value":"54REBYJ7ZYIY","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"54REBYJ7ZYIY3WA5","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"54REBYJ7","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:dd133032107c3b0484b4e4fcb363dcaf7ddbb9f0edded47403ff678253d7394b","target":"graph","created_at":"2026-05-18T01:31:12Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $G$. It follows that any isometric action of a random group of the graph ","authors_text":"Tetsu Toyoda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-03T16:46:54Z","title":"Fixed point property for a CAT(0) space which admits a proper cocompact group action"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0729","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9fd632b2b499e940fca525cb6ec90c9cc414f52c4f03f62997224fa42d0198e7","target":"record","created_at":"2026-05-18T01:31:12Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"dc0edd6e17550369982399aaa400563c58e853f3d1916fddb8160f7028f0aa80","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-03T16:46:54Z","title_canon_sha256":"af1e0f34a742ed8f63e549516ba44477a747e159f4a44e4a277b55e0e7906f62"},"schema_version":"1.0","source":{"id":"1102.0729","kind":"arxiv","version":4}},"canonical_sha256":"ef2240e13fce118dd81dafdffae268478eb0ea23a55073bb79641df5233d9791","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ef2240e13fce118dd81dafdffae268478eb0ea23a55073bb79641df5233d9791","first_computed_at":"2026-05-18T01:31:12.174027Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:12.174027Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HJjzxR+BnqZUezs17FYoxeKG+TXP7Ontp+Kg8kxl55357qJB0MpGQ4AEwtjlec1JEf+El5oTuwqTborm+SwVAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:12.174671Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.0729","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9fd632b2b499e940fca525cb6ec90c9cc414f52c4f03f62997224fa42d0198e7","sha256:dd133032107c3b0484b4e4fcb363dcaf7ddbb9f0edded47403ff678253d7394b"],"state_sha256":"83a020a42c2a7fef3890092fd2c81c8b1395bfd9d93e7f31f92ed42cc686b51c"}