{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:P4M7TMW7LG2GFBT6NHYIKZM2X6","short_pith_number":"pith:P4M7TMW7","canonical_record":{"source":{"id":"1609.06698","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-21T19:41:12Z","cross_cats_sorted":["math.GR","math.MG"],"title_canon_sha256":"e8aeab625871de3f69ebd8e4e2d5256530fc8ed9d7e144d226be25c24a47cc86","abstract_canon_sha256":"ca0cb92c6210692e25163084353ef28b3d6224256df1c00bdf355655215f4bbf"},"schema_version":"1.0"},"canonical_sha256":"7f19f9b2df59b462867e69f085659abfb9fcd751d027503dfd511dd427d0e976","source":{"kind":"arxiv","id":"1609.06698","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.06698","created_at":"2026-05-18T00:34:52Z"},{"alias_kind":"arxiv_version","alias_value":"1609.06698v2","created_at":"2026-05-18T00:34:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06698","created_at":"2026-05-18T00:34:52Z"},{"alias_kind":"pith_short_12","alias_value":"P4M7TMW7LG2G","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"P4M7TMW7LG2GFBT6","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"P4M7TMW7","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:P4M7TMW7LG2GFBT6NHYIKZM2X6","target":"record","payload":{"canonical_record":{"source":{"id":"1609.06698","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-21T19:41:12Z","cross_cats_sorted":["math.GR","math.MG"],"title_canon_sha256":"e8aeab625871de3f69ebd8e4e2d5256530fc8ed9d7e144d226be25c24a47cc86","abstract_canon_sha256":"ca0cb92c6210692e25163084353ef28b3d6224256df1c00bdf355655215f4bbf"},"schema_version":"1.0"},"canonical_sha256":"7f19f9b2df59b462867e69f085659abfb9fcd751d027503dfd511dd427d0e976","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:52.665641Z","signature_b64":"pYPVQID918TpYoqJ/Wm0zG7j9UdmeKDO3NL/FH01k80odvOskg6QjeTxAkn9dJRXBIToPp/Udj34gtbGyjhtBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f19f9b2df59b462867e69f085659abfb9fcd751d027503dfd511dd427d0e976","last_reissued_at":"2026-05-18T00:34:52.664772Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:52.664772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.06698","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":"2026-05-18T00:34:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vrNaBqlhASpwhlJbmCVuPjgRIoHecDmbXs5wsVjl+UFET7B0yk+nvOYV55CLk1Te1cGO/KYQ/WYrAT2NlvXqBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:43:46.893629Z"},"content_sha256":"4df2181fbd9251bd642ca010127d6dce9633837cd023932e18477621b1f083b0","schema_version":"1.0","event_id":"sha256:4df2181fbd9251bd642ca010127d6dce9633837cd023932e18477621b1f083b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:P4M7TMW7LG2GFBT6NHYIKZM2X6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.MG"],"primary_cat":"math.GT","authors_text":"Matthew Gentry Durham, Samuel J. Taylor, Tarik Aougab","submitted_at":"2016-09-21T19:41:12Z","abstract_excerpt":"We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06698","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":"2026-05-18T00:34:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hY/TYcs5AEiXVfpaQ1RF1RSnR+Fb9yUfEoeGnA6PbUX23pn+9b1Gw2vCithkjRrtZcHBGQjcZYK6gLsEbIqJDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:43:46.894293Z"},"content_sha256":"7dc35d253e88c1d1ff7d60904b7a5e5c6f5091cced771e18684069c47a4ba332","schema_version":"1.0","event_id":"sha256:7dc35d253e88c1d1ff7d60904b7a5e5c6f5091cced771e18684069c47a4ba332"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6/bundle.json","state_url":"https://pith.science/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6/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":"2026-05-25T17:43:46Z","links":{"resolver":"https://pith.science/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6","bundle":"https://pith.science/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6/bundle.json","state":"https://pith.science/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P4M7TMW7LG2GFBT6NHYIKZM2X6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:P4M7TMW7LG2GFBT6NHYIKZM2X6","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":"ca0cb92c6210692e25163084353ef28b3d6224256df1c00bdf355655215f4bbf","cross_cats_sorted":["math.GR","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-21T19:41:12Z","title_canon_sha256":"e8aeab625871de3f69ebd8e4e2d5256530fc8ed9d7e144d226be25c24a47cc86"},"schema_version":"1.0","source":{"id":"1609.06698","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.06698","created_at":"2026-05-18T00:34:52Z"},{"alias_kind":"arxiv_version","alias_value":"1609.06698v2","created_at":"2026-05-18T00:34:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06698","created_at":"2026-05-18T00:34:52Z"},{"alias_kind":"pith_short_12","alias_value":"P4M7TMW7LG2G","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"P4M7TMW7LG2GFBT6","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"P4M7TMW7","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:7dc35d253e88c1d1ff7d60904b7a5e5c6f5091cced771e18684069c47a4ba332","target":"graph","created_at":"2026-05-18T00:34:52Z","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 stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.","authors_text":"Matthew Gentry Durham, Samuel J. Taylor, Tarik Aougab","cross_cats":["math.GR","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-21T19:41:12Z","title":"Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06698","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":"sha256:4df2181fbd9251bd642ca010127d6dce9633837cd023932e18477621b1f083b0","target":"record","created_at":"2026-05-18T00:34:52Z","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":"ca0cb92c6210692e25163084353ef28b3d6224256df1c00bdf355655215f4bbf","cross_cats_sorted":["math.GR","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-09-21T19:41:12Z","title_canon_sha256":"e8aeab625871de3f69ebd8e4e2d5256530fc8ed9d7e144d226be25c24a47cc86"},"schema_version":"1.0","source":{"id":"1609.06698","kind":"arxiv","version":2}},"canonical_sha256":"7f19f9b2df59b462867e69f085659abfb9fcd751d027503dfd511dd427d0e976","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f19f9b2df59b462867e69f085659abfb9fcd751d027503dfd511dd427d0e976","first_computed_at":"2026-05-18T00:34:52.664772Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:52.664772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pYPVQID918TpYoqJ/Wm0zG7j9UdmeKDO3NL/FH01k80odvOskg6QjeTxAkn9dJRXBIToPp/Udj34gtbGyjhtBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:52.665641Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.06698","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4df2181fbd9251bd642ca010127d6dce9633837cd023932e18477621b1f083b0","sha256:7dc35d253e88c1d1ff7d60904b7a5e5c6f5091cced771e18684069c47a4ba332"],"state_sha256":"0acbad3dc889f9dc727cc29575e8a355ddd4bde7fae2a05ce9afa580afb0805f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LAftDyuujwuJtbd8Sns7ecMyNCUU4dty0r1vBCGi5WkwVlgFKe27FVVQ69PlJ2DeHgR++ziwGSGr/NHn2g91Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T17:43:46.897574Z","bundle_sha256":"6b1275688d93e067fe0d34556b6339b1de8a59d59f3fb01fb00fada67a9f8d23"}}