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We determine a family of quasi-isometry invariants for such $\\Gamma$, namely the $k$-dimensional Dehn functions, which measure the difficulty to fill $k$-spheres by $(k+1)$-balls (for $1\\leq k\\leq \\dim\\ X-1$). Since the group $\\Gamma$ is quasi-isometric to the associated CAT(0) space $X$, assertions about Dehn functions for $\\Gamma$ are equivalent tothe correspo"},"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":"1205.4923","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-05-22T14:23:44Z","cross_cats_sorted":["math.GR","math.MG"],"title_canon_sha256":"9c09370008150d4f5af39eac09f04a6b6760832e3ca802d5385be1e996059510","abstract_canon_sha256":"1c91b75563cc488a677a4e95670c36f35e875dee14dd214e97adcda5fbeadd53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:11.313623Z","signature_b64":"8+Pzs8XR9XTp6dAqbgVuyUvrTtlC5XjiA93sh8ndPCgqeswGY5u3UdiLmbzHr1tBC1a5huD9DcKaYV8836oUBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc666bbdb69acebe96e92d6982fda55e3d56ad5da45ed1af3e175d3a9b1a28a9","last_reissued_at":"2026-05-18T03:55:11.312907Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:11.312907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal higher-dimensional Dehn functions for some CAT(0) lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.MG"],"primary_cat":"math.DG","authors_text":"Enrico Leuzinger","submitted_at":"2012-05-22T14:23:44Z","abstract_excerpt":"Let $X=S\\times E \\times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. 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Since the group $\\Gamma$ is quasi-isometric to the associated CAT(0) space $X$, assertions about Dehn functions for $\\Gamma$ are equivalent tothe correspo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4923","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1205.4923","created_at":"2026-05-18T03:55:11.313005+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.4923v1","created_at":"2026-05-18T03:55:11.313005+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4923","created_at":"2026-05-18T03:55:11.313005+00:00"},{"alias_kind":"pith_short_12","alias_value":"7RTGXPNWTLHL","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"7RTGXPNWTLHL5FXJ","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"7RTGXPNW","created_at":"2026-05-18T12:26:58.693483+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY","json":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY.json","graph_json":"https://pith.science/api/pith-number/7RTGXPNWTLHL5FXJFVUYF7NFLY/graph.json","events_json":"https://pith.science/api/pith-number/7RTGXPNWTLHL5FXJFVUYF7NFLY/events.json","paper":"https://pith.science/paper/7RTGXPNW"},"agent_actions":{"view_html":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY","download_json":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY.json","view_paper":"https://pith.science/paper/7RTGXPNW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.4923&json=true","fetch_graph":"https://pith.science/api/pith-number/7RTGXPNWTLHL5FXJFVUYF7NFLY/graph.json","fetch_events":"https://pith.science/api/pith-number/7RTGXPNWTLHL5FXJFVUYF7NFLY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/action/storage_attestation","attest_author":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/action/author_attestation","sign_citation":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/action/citation_signature","submit_replication":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/action/replication_record"}},"created_at":"2026-05-18T03:55:11.313005+00:00","updated_at":"2026-05-18T03:55:11.313005+00:00"}