{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:VQU2Y73O2TJEGIVVORXG2BVYDQ","short_pith_number":"pith:VQU2Y73O","canonical_record":{"source":{"id":"1601.00101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-01T18:15:58Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"a812af9671891147ac729a74390361f123fdef464f8883b66fbcd5387eb282bf","abstract_canon_sha256":"3c2c0afe2bcb04bccfdac27a2be55c236642e482b1bcfcb711f31301333a381c"},"schema_version":"1.0"},"canonical_sha256":"ac29ac7f6ed4d24322b5746e6d06b81c0725634cc4ecb1669337e816fafb5d3e","source":{"kind":"arxiv","id":"1601.00101","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00101","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00101v2","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00101","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"pith_short_12","alias_value":"VQU2Y73O2TJE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VQU2Y73O2TJEGIVV","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VQU2Y73O","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:VQU2Y73O2TJEGIVVORXG2BVYDQ","target":"record","payload":{"canonical_record":{"source":{"id":"1601.00101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-01T18:15:58Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"a812af9671891147ac729a74390361f123fdef464f8883b66fbcd5387eb282bf","abstract_canon_sha256":"3c2c0afe2bcb04bccfdac27a2be55c236642e482b1bcfcb711f31301333a381c"},"schema_version":"1.0"},"canonical_sha256":"ac29ac7f6ed4d24322b5746e6d06b81c0725634cc4ecb1669337e816fafb5d3e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:08.479665Z","signature_b64":"8R0kIdUIiOgCVIDGU74xsEh/DEcfmrW3jnBvrozCHVW095DSC/yT9ValQ72XtoXFR4dQ5g911sWV639nBA9WBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac29ac7f6ed4d24322b5746e6d06b81c0725634cc4ecb1669337e816fafb5d3e","last_reissued_at":"2026-05-18T00:45:08.479142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:08.479142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.00101","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":"2026-05-18T00:45:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lXq1W1El5TBa5uN7P1KthklUeMOJnXubQUX2M5NRXHpnsAbndzMxXgmqlcRN7rKJ8U71VfHmMt95yJcXPXVIAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:04:16.552792Z"},"content_sha256":"8202d8987ee394f3a32b0e0c153003b099a66e63a09a9e5437bed4cd11eab44d","schema_version":"1.0","event_id":"sha256:8202d8987ee394f3a32b0e0c153003b099a66e63a09a9e5437bed4cd11eab44d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:VQU2Y73O2TJEGIVVORXG2BVYDQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The co-surface graph and the geometry of hyperbolic free group extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Samuel J. Taylor, Spencer Dowdall","submitted_at":"2016-01-01T18:15:58Z","abstract_excerpt":"We introduce the co-surface graph $\\mathcal{CS}$ of a finitely generated free group $\\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\\mathbb{F}$-trees and use this to prove that a finitely generated subgroup of $\\mathrm{Out}(\\mathbb{F})$ quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00101","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":"2026-05-18T00:45:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WsVeVKo97ESb4kr34R5OaNC6DlxqGO1xpeqRCbWEPJo2c+QmEDI7SVfnqdIEWBVlOVFrgQzUbfghSJIbteTZBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:04:16.553462Z"},"content_sha256":"dd3b07b68f3bc486a7fc2e4ba6b450dea08bf90d3b9e94da10926bf652a299af","schema_version":"1.0","event_id":"sha256:dd3b07b68f3bc486a7fc2e4ba6b450dea08bf90d3b9e94da10926bf652a299af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ/bundle.json","state_url":"https://pith.science/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ/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":"2026-05-25T23:04:16Z","links":{"resolver":"https://pith.science/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ","bundle":"https://pith.science/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ/bundle.json","state":"https://pith.science/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VQU2Y73O2TJEGIVVORXG2BVYDQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VQU2Y73O2TJEGIVVORXG2BVYDQ","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":"3c2c0afe2bcb04bccfdac27a2be55c236642e482b1bcfcb711f31301333a381c","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-01T18:15:58Z","title_canon_sha256":"a812af9671891147ac729a74390361f123fdef464f8883b66fbcd5387eb282bf"},"schema_version":"1.0","source":{"id":"1601.00101","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00101","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00101v2","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00101","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"pith_short_12","alias_value":"VQU2Y73O2TJE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VQU2Y73O2TJEGIVV","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VQU2Y73O","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:dd3b07b68f3bc486a7fc2e4ba6b450dea08bf90d3b9e94da10926bf652a299af","target":"graph","created_at":"2026-05-18T00:45:08Z","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 introduce the co-surface graph $\\mathcal{CS}$ of a finitely generated free group $\\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\\mathbb{F}$-trees and use this to prove that a finitely generated subgroup of $\\mathrm{Out}(\\mathbb{F})$ quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of","authors_text":"Samuel J. Taylor, Spencer Dowdall","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-01T18:15:58Z","title":"The co-surface graph and the geometry of hyperbolic free group extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00101","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":"sha256:8202d8987ee394f3a32b0e0c153003b099a66e63a09a9e5437bed4cd11eab44d","target":"record","created_at":"2026-05-18T00:45:08Z","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":"3c2c0afe2bcb04bccfdac27a2be55c236642e482b1bcfcb711f31301333a381c","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-01T18:15:58Z","title_canon_sha256":"a812af9671891147ac729a74390361f123fdef464f8883b66fbcd5387eb282bf"},"schema_version":"1.0","source":{"id":"1601.00101","kind":"arxiv","version":2}},"canonical_sha256":"ac29ac7f6ed4d24322b5746e6d06b81c0725634cc4ecb1669337e816fafb5d3e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac29ac7f6ed4d24322b5746e6d06b81c0725634cc4ecb1669337e816fafb5d3e","first_computed_at":"2026-05-18T00:45:08.479142Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:08.479142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8R0kIdUIiOgCVIDGU74xsEh/DEcfmrW3jnBvrozCHVW095DSC/yT9ValQ72XtoXFR4dQ5g911sWV639nBA9WBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:08.479665Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.00101","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8202d8987ee394f3a32b0e0c153003b099a66e63a09a9e5437bed4cd11eab44d","sha256:dd3b07b68f3bc486a7fc2e4ba6b450dea08bf90d3b9e94da10926bf652a299af"],"state_sha256":"b21aa058b2011ec922ac42096d97e7bb9a3823bda9f80f5511d2e389aba189c4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EOXQYloDlrDU1/yo5yTQCn8mSoBHfjFnwUrSUQo8FguvDHUT9McD2g50I9+7zKenUf014g3FhwRCoDv0Vfn8CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T23:04:16.557030Z","bundle_sha256":"8bbce1ec0712e70c236700fb60d0c54cf6614a2c9bb3395332c918dd04da52e2"}}