{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:LNJDCNIWG6COU24WZNA6FZNSF5","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":"96f71ad86701779fc6cd793107c0532166c42f0991695f2ee4bc52f6e0fa6b0a","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-04T22:52:58Z","title_canon_sha256":"29ce4ce1e8135ad84a3d028756d507d131a6fb9ce981ce1fd64ea3a2143dd9bd"},"schema_version":"1.0","source":{"id":"1002.1096","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.1096","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"arxiv_version","alias_value":"1002.1096v1","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.1096","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"pith_short_12","alias_value":"LNJDCNIWG6CO","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LNJDCNIWG6COU24W","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LNJDCNIW","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:f26d3425c71f0e5cebb6c899d2f780d4c422a6885a39ca026cd621109074c9c5","target":"graph","created_at":"2026-05-18T02:41:50Z","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":"Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $(G,H)$ acts on $(X, Y)$, we define the $k$-volume distortion function of $H$ in $G$ to measure the large-scale difference between the volumes of such fillings. We show that these functions are quasi-isometry invariants, and thus independent of the choice of spaces, and provide several bounds in terms of other group properties, such as Dehn functions. We also compute the volume distortion in a number of examples, includin","authors_text":"Hanna Bennett","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-04T22:52:58Z","title":"Volume distortion in groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1096","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:a3b6276f0251e239f86034bf08358e6105820aaf2315ad629c56cd8615a862a7","target":"record","created_at":"2026-05-18T02:41:50Z","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":"96f71ad86701779fc6cd793107c0532166c42f0991695f2ee4bc52f6e0fa6b0a","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-04T22:52:58Z","title_canon_sha256":"29ce4ce1e8135ad84a3d028756d507d131a6fb9ce981ce1fd64ea3a2143dd9bd"},"schema_version":"1.0","source":{"id":"1002.1096","kind":"arxiv","version":1}},"canonical_sha256":"5b523135163784ea6b96cb41e2e5b22f6c5beaa8c31a15916d8ba93b4ee06560","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b523135163784ea6b96cb41e2e5b22f6c5beaa8c31a15916d8ba93b4ee06560","first_computed_at":"2026-05-18T02:41:50.965783Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:50.965783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7/lfYpy7UBeKO8QEnw7cUQxhpAva+ul0ayrpcrLdVTt5qLI1So/rNLFCMaMxHqtC39oS2IoAqeJZvSYajv9sBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:50.966273Z","signed_message":"canonical_sha256_bytes"},"source_id":"1002.1096","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a3b6276f0251e239f86034bf08358e6105820aaf2315ad629c56cd8615a862a7","sha256:f26d3425c71f0e5cebb6c899d2f780d4c422a6885a39ca026cd621109074c9c5"],"state_sha256":"b3e0c350a1d1aded8d992e8d20b6f86f0c3ee274fabbf27ea10993c9fb718138"}