{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7RTGXPNWTLHL5FXJFVUYF7NFLY","short_pith_number":"pith:7RTGXPNW","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"},"canonical_sha256":"fc666bbdb69acebe96e92d6982fda55e3d56ad5da45ed1af3e175d3a9b1a28a9","source":{"kind":"arxiv","id":"1205.4923","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4923","created_at":"2026-05-18T03:55:11Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4923v1","created_at":"2026-05-18T03:55:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4923","created_at":"2026-05-18T03:55:11Z"},{"alias_kind":"pith_short_12","alias_value":"7RTGXPNWTLHL","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7RTGXPNWTLHL5FXJ","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7RTGXPNW","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7RTGXPNWTLHL5FXJFVUYF7NFLY","target":"record","payload":{"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"},"canonical_sha256":"fc666bbdb69acebe96e92d6982fda55e3d56ad5da45ed1af3e175d3a9b1a28a9","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"},"source_kind":"arxiv","source_id":"1205.4923","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:55:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8RSnVzJ1iKogJrU24eaWjrB80xY8dVPaaIK06jdUSeCsZSf5h0F/okE9n1Wsq4QaA6s0XeE+Qu3DYBcIIexjAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:39:48.085418Z"},"content_sha256":"b78a0b8c5643b2b92a5cd1b327f2f5fa33c6a2c4065c70faea982e8c11ba5d51","schema_version":"1.0","event_id":"sha256:b78a0b8c5643b2b92a5cd1b327f2f5fa33c6a2c4065c70faea982e8c11ba5d51"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7RTGXPNWTLHL5FXJFVUYF7NFLY","target":"graph","payload":{"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. Let $\\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. 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"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:55:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0em+BwCh/zykC85+cMgTyho2Q2dq3KdWo0HHugIpW6fKkcDOM7MF1AibhIP2zkp70ylInn1zyKqULJhWOyhFBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:39:48.086065Z"},"content_sha256":"672c65ca46c17ff72f2c20816681a76ae2a567c487d13da938e2bb157209edc2","schema_version":"1.0","event_id":"sha256:672c65ca46c17ff72f2c20816681a76ae2a567c487d13da938e2bb157209edc2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/bundle.json","state_url":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T08:39:48Z","links":{"resolver":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY","bundle":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/bundle.json","state":"https://pith.science/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7RTGXPNWTLHL5FXJFVUYF7NFLY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7RTGXPNWTLHL5FXJFVUYF7NFLY","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":"1c91b75563cc488a677a4e95670c36f35e875dee14dd214e97adcda5fbeadd53","cross_cats_sorted":["math.GR","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-05-22T14:23:44Z","title_canon_sha256":"9c09370008150d4f5af39eac09f04a6b6760832e3ca802d5385be1e996059510"},"schema_version":"1.0","source":{"id":"1205.4923","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4923","created_at":"2026-05-18T03:55:11Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4923v1","created_at":"2026-05-18T03:55:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4923","created_at":"2026-05-18T03:55:11Z"},{"alias_kind":"pith_short_12","alias_value":"7RTGXPNWTLHL","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7RTGXPNWTLHL5FXJ","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7RTGXPNW","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:672c65ca46c17ff72f2c20816681a76ae2a567c487d13da938e2bb157209edc2","target":"graph","created_at":"2026-05-18T03:55:11Z","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":"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. Let $\\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. 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","authors_text":"Enrico Leuzinger","cross_cats":["math.GR","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-05-22T14:23:44Z","title":"Optimal higher-dimensional Dehn functions for some CAT(0) lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4923","kind":"arxiv","version":1},"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:b78a0b8c5643b2b92a5cd1b327f2f5fa33c6a2c4065c70faea982e8c11ba5d51","target":"record","created_at":"2026-05-18T03:55:11Z","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":"1c91b75563cc488a677a4e95670c36f35e875dee14dd214e97adcda5fbeadd53","cross_cats_sorted":["math.GR","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-05-22T14:23:44Z","title_canon_sha256":"9c09370008150d4f5af39eac09f04a6b6760832e3ca802d5385be1e996059510"},"schema_version":"1.0","source":{"id":"1205.4923","kind":"arxiv","version":1}},"canonical_sha256":"fc666bbdb69acebe96e92d6982fda55e3d56ad5da45ed1af3e175d3a9b1a28a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc666bbdb69acebe96e92d6982fda55e3d56ad5da45ed1af3e175d3a9b1a28a9","first_computed_at":"2026-05-18T03:55:11.312907Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:11.312907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8+Pzs8XR9XTp6dAqbgVuyUvrTtlC5XjiA93sh8ndPCgqeswGY5u3UdiLmbzHr1tBC1a5huD9DcKaYV8836oUBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:11.313623Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.4923","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b78a0b8c5643b2b92a5cd1b327f2f5fa33c6a2c4065c70faea982e8c11ba5d51","sha256:672c65ca46c17ff72f2c20816681a76ae2a567c487d13da938e2bb157209edc2"],"state_sha256":"ca66a1a3527cf58721ccda27599e0f2ffd577367d245ffd0fe20f62e0d16e533"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kQVUErNPGfiPegtYy4c7VD6vUePUldLq2yPZGNng8fbIHgs3IzG04oJvvoCY1LLaoCKGr+/npBpd7l6/pI/ZBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T08:39:48.089787Z","bundle_sha256":"2fbdc0bf82ef0d2e4f01c807a90f13a571720f02c73a969ce267444919d5b83d"}}