{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:X4J52PZPW3V3BFXZWXG7WDJVKU","short_pith_number":"pith:X4J52PZP","canonical_record":{"source":{"id":"1511.09448","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-11-30T20:01:43Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"822b9cf0333fb7aaaa2b7d7628fdd5badf2785f5b7136606d4e6a93e8b69f248","abstract_canon_sha256":"44a8ad1517c977a9a9f5f80a5250cc74afe6bd37cef2d684fb7d45d2b4a3cf7b"},"schema_version":"1.0"},"canonical_sha256":"bf13dd3f2fb6ebb096f9b5cdfb0d35551bc70c4ae1ccacbebc1f39fd2af172cb","source":{"kind":"arxiv","id":"1511.09448","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.09448","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"arxiv_version","alias_value":"1511.09448v2","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09448","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"pith_short_12","alias_value":"X4J52PZPW3V3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X4J52PZPW3V3BFXZ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X4J52PZP","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:X4J52PZPW3V3BFXZWXG7WDJVKU","target":"record","payload":{"canonical_record":{"source":{"id":"1511.09448","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-11-30T20:01:43Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"822b9cf0333fb7aaaa2b7d7628fdd5badf2785f5b7136606d4e6a93e8b69f248","abstract_canon_sha256":"44a8ad1517c977a9a9f5f80a5250cc74afe6bd37cef2d684fb7d45d2b4a3cf7b"},"schema_version":"1.0"},"canonical_sha256":"bf13dd3f2fb6ebb096f9b5cdfb0d35551bc70c4ae1ccacbebc1f39fd2af172cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:42.205233Z","signature_b64":"BFiyf0B9Ag7T0KcLq7sLD8JaaAcTfkt2KCJdTCCQzGxQP8fQYzpM5WQTQQB34gBLL1jAVeIQvTJsaWG+XAP1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf13dd3f2fb6ebb096f9b5cdfb0d35551bc70c4ae1ccacbebc1f39fd2af172cb","last_reissued_at":"2026-05-18T01:01:42.204694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:42.204694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.09448","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kkOC9ahAEYPOMejBFKOgz2UzensVOrkyNs0bIisF363B0cDoYZd646ZZawrobU8PSwwuDktEMMZK2VvCBHubBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T18:05:41.921786Z"},"content_sha256":"5798dd487746a8c93c50bf7e9b6cd0b6d52df43a90ea274f9252cf90c7ce5bdc","schema_version":"1.0","event_id":"sha256:5798dd487746a8c93c50bf7e9b6cd0b6d52df43a90ea274f9252cf90c7ce5bdc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:X4J52PZPW3V3BFXZWXG7WDJVKU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Volume and non-existence of compact Clifford-Klein forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Nicolas Tholozan","submitted_at":"2015-11-30T20:01:43Z","abstract_excerpt":"This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $\\Gamma$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that the volume of $\\Gamma \\backslash G/H$ is the integral, over a certain homology class of $\\Gamma$, of a $G$-invariant form on $G/K$ (where $K$ is a maximal compact subgroup of $G$).\n  As a corollary, we obtain a large class of homogeneous spaces the compact quotients of which have rational volume. For instance, compact quotients of pseudo-Riemannian spaces o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09448","kind":"arxiv","version":2},"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":"2026-05-18T01:01:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"trVX8fXHeGidH9ORVN+ylgk6UIQm10N3nIJZLdw6iSyNCWpBtTQWKkvKu+d7MAAWPpR5bo/kOuTprOLcD3yPCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T18:05:41.922351Z"},"content_sha256":"dbe197db1677974db4e5111b2fdf5dc1ff1b2d7215ecf03695cf3fdae20777df","schema_version":"1.0","event_id":"sha256:dbe197db1677974db4e5111b2fdf5dc1ff1b2d7215ecf03695cf3fdae20777df"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X4J52PZPW3V3BFXZWXG7WDJVKU/bundle.json","state_url":"https://pith.science/pith/X4J52PZPW3V3BFXZWXG7WDJVKU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X4J52PZPW3V3BFXZWXG7WDJVKU/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":"2026-05-21T18:05:41Z","links":{"resolver":"https://pith.science/pith/X4J52PZPW3V3BFXZWXG7WDJVKU","bundle":"https://pith.science/pith/X4J52PZPW3V3BFXZWXG7WDJVKU/bundle.json","state":"https://pith.science/pith/X4J52PZPW3V3BFXZWXG7WDJVKU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X4J52PZPW3V3BFXZWXG7WDJVKU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X4J52PZPW3V3BFXZWXG7WDJVKU","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":"44a8ad1517c977a9a9f5f80a5250cc74afe6bd37cef2d684fb7d45d2b4a3cf7b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-11-30T20:01:43Z","title_canon_sha256":"822b9cf0333fb7aaaa2b7d7628fdd5badf2785f5b7136606d4e6a93e8b69f248"},"schema_version":"1.0","source":{"id":"1511.09448","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.09448","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"arxiv_version","alias_value":"1511.09448v2","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09448","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"pith_short_12","alias_value":"X4J52PZPW3V3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X4J52PZPW3V3BFXZ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X4J52PZP","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:dbe197db1677974db4e5111b2fdf5dc1ff1b2d7215ecf03695cf3fdae20777df","target":"graph","created_at":"2026-05-18T01:01:42Z","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":"This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $\\Gamma$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that the volume of $\\Gamma \\backslash G/H$ is the integral, over a certain homology class of $\\Gamma$, of a $G$-invariant form on $G/K$ (where $K$ is a maximal compact subgroup of $G$).\n  As a corollary, we obtain a large class of homogeneous spaces the compact quotients of which have rational volume. For instance, compact quotients of pseudo-Riemannian spaces o","authors_text":"Nicolas Tholozan","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-11-30T20:01:43Z","title":"Volume and non-existence of compact Clifford-Klein forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09448","kind":"arxiv","version":2},"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":"sha256:5798dd487746a8c93c50bf7e9b6cd0b6d52df43a90ea274f9252cf90c7ce5bdc","target":"record","created_at":"2026-05-18T01:01:42Z","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":"44a8ad1517c977a9a9f5f80a5250cc74afe6bd37cef2d684fb7d45d2b4a3cf7b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-11-30T20:01:43Z","title_canon_sha256":"822b9cf0333fb7aaaa2b7d7628fdd5badf2785f5b7136606d4e6a93e8b69f248"},"schema_version":"1.0","source":{"id":"1511.09448","kind":"arxiv","version":2}},"canonical_sha256":"bf13dd3f2fb6ebb096f9b5cdfb0d35551bc70c4ae1ccacbebc1f39fd2af172cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf13dd3f2fb6ebb096f9b5cdfb0d35551bc70c4ae1ccacbebc1f39fd2af172cb","first_computed_at":"2026-05-18T01:01:42.204694Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:42.204694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BFiyf0B9Ag7T0KcLq7sLD8JaaAcTfkt2KCJdTCCQzGxQP8fQYzpM5WQTQQB34gBLL1jAVeIQvTJsaWG+XAP1Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:42.205233Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.09448","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5798dd487746a8c93c50bf7e9b6cd0b6d52df43a90ea274f9252cf90c7ce5bdc","sha256:dbe197db1677974db4e5111b2fdf5dc1ff1b2d7215ecf03695cf3fdae20777df"],"state_sha256":"0ece106c6b50bd2fe241372a3fe33fb69a6e92bc5a128c29bf4ea6e99a2ff787"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eW0pqBVMckZNb82GCozyA6i2VC/XhUVccENen8jyqmOCm7/wwCD9NB7qS204WdImeqGHoBXJPCODbg0L18xLBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T18:05:41.925440Z","bundle_sha256":"ace89a60a8b24686112656b4c4fa7a9b826cd93b7cb499f4583ef3cb6d0c65b3"}}