{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:FOHMQJR54Y2F32YZGVUVAP6UJJ","short_pith_number":"pith:FOHMQJR5","canonical_record":{"source":{"id":"1501.04079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-16T19:09:06Z","cross_cats_sorted":["math.GR","math.LO"],"title_canon_sha256":"cb6b73d6a65f85d155b38928aebbbd42cc3b4bfca1797ca44f64e866b5853e5f","abstract_canon_sha256":"c76bbe66e810ba3cd592950e7d64e283008da30e7080c7375cdf073edcd599aa"},"schema_version":"1.0"},"canonical_sha256":"2b8ec8263de6345deb193569503fd44a6d070d1705b073049ceb3ad528089ab2","source":{"kind":"arxiv","id":"1501.04079","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.04079","created_at":"2026-05-18T01:23:24Z"},{"alias_kind":"arxiv_version","alias_value":"1501.04079v2","created_at":"2026-05-18T01:23:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.04079","created_at":"2026-05-18T01:23:24Z"},{"alias_kind":"pith_short_12","alias_value":"FOHMQJR54Y2F","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FOHMQJR54Y2F32YZ","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FOHMQJR5","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:FOHMQJR54Y2F32YZGVUVAP6UJJ","target":"record","payload":{"canonical_record":{"source":{"id":"1501.04079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-16T19:09:06Z","cross_cats_sorted":["math.GR","math.LO"],"title_canon_sha256":"cb6b73d6a65f85d155b38928aebbbd42cc3b4bfca1797ca44f64e866b5853e5f","abstract_canon_sha256":"c76bbe66e810ba3cd592950e7d64e283008da30e7080c7375cdf073edcd599aa"},"schema_version":"1.0"},"canonical_sha256":"2b8ec8263de6345deb193569503fd44a6d070d1705b073049ceb3ad528089ab2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:24.571183Z","signature_b64":"ykfuyNVqTm2Q4OvBfnVSdupXgD8eRMNrDoueHSXMPLhzvm4VpYxE6k2e+RaBmi5J1hxShZHbL853A2M23gfxDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b8ec8263de6345deb193569503fd44a6d070d1705b073049ceb3ad528089ab2","last_reissued_at":"2026-05-18T01:23:24.570598Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:24.570598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.04079","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":"2026-05-18T01:23:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cxKraABazI7Osh64vVSJPvCJI7sLs6Mxx0BaCqfrW+GMHKbOZVH1wX/iI/QwqXm0f3bsnmp/FROiuBPwaJaKBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T14:33:31.905621Z"},"content_sha256":"f4a013f0b92dec97a390ddcaa3cbf7e41c89f3220f03ae1a6d053284e19883b9","schema_version":"1.0","event_id":"sha256:f4a013f0b92dec97a390ddcaa3cbf7e41c89f3220f03ae1a6d053284e19883b9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:FOHMQJR54Y2F32YZGVUVAP6UJJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topology and convexity in the space of actions modulo weak equivalence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.LO"],"primary_cat":"math.DS","authors_text":"Peter Burton","submitted_at":"2015-01-16T19:09:06Z","abstract_excerpt":"We analyse the structure of the quotient $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We show that the convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is compatible with the topology, and as a consequence deduce that $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ for amenable $\\Gamma$ by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04079","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":"2026-05-18T01:23:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"96iNQtL7CPNN/cdrnaBLmi7T8qrCIxUMtgq9vReqIn48143flE4Y3kdF7IsIDDtJbG6OWLCcZqEr+M/mCs/VCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T14:33:31.905976Z"},"content_sha256":"02b772bf7080b28ffa6c3e7ccde827acc5b044422a579cc321b32e150c0ea5c9","schema_version":"1.0","event_id":"sha256:02b772bf7080b28ffa6c3e7ccde827acc5b044422a579cc321b32e150c0ea5c9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ/bundle.json","state_url":"https://pith.science/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ/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":"2026-05-30T14:33:31Z","links":{"resolver":"https://pith.science/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ","bundle":"https://pith.science/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ/bundle.json","state":"https://pith.science/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FOHMQJR54Y2F32YZGVUVAP6UJJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FOHMQJR54Y2F32YZGVUVAP6UJJ","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":"c76bbe66e810ba3cd592950e7d64e283008da30e7080c7375cdf073edcd599aa","cross_cats_sorted":["math.GR","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-16T19:09:06Z","title_canon_sha256":"cb6b73d6a65f85d155b38928aebbbd42cc3b4bfca1797ca44f64e866b5853e5f"},"schema_version":"1.0","source":{"id":"1501.04079","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.04079","created_at":"2026-05-18T01:23:24Z"},{"alias_kind":"arxiv_version","alias_value":"1501.04079v2","created_at":"2026-05-18T01:23:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.04079","created_at":"2026-05-18T01:23:24Z"},{"alias_kind":"pith_short_12","alias_value":"FOHMQJR54Y2F","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FOHMQJR54Y2F32YZ","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FOHMQJR5","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:02b772bf7080b28ffa6c3e7ccde827acc5b044422a579cc321b32e150c0ea5c9","target":"graph","created_at":"2026-05-18T01:23:24Z","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 analyse the structure of the quotient $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We show that the convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is compatible with the topology, and as a consequence deduce that $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of $\\mathrm{A}_\\sim(\\Gamma,X,\\mu)$ for amenable $\\Gamma$ by","authors_text":"Peter Burton","cross_cats":["math.GR","math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-16T19:09:06Z","title":"Topology and convexity in the space of actions modulo weak equivalence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04079","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:f4a013f0b92dec97a390ddcaa3cbf7e41c89f3220f03ae1a6d053284e19883b9","target":"record","created_at":"2026-05-18T01:23:24Z","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":"c76bbe66e810ba3cd592950e7d64e283008da30e7080c7375cdf073edcd599aa","cross_cats_sorted":["math.GR","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-16T19:09:06Z","title_canon_sha256":"cb6b73d6a65f85d155b38928aebbbd42cc3b4bfca1797ca44f64e866b5853e5f"},"schema_version":"1.0","source":{"id":"1501.04079","kind":"arxiv","version":2}},"canonical_sha256":"2b8ec8263de6345deb193569503fd44a6d070d1705b073049ceb3ad528089ab2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b8ec8263de6345deb193569503fd44a6d070d1705b073049ceb3ad528089ab2","first_computed_at":"2026-05-18T01:23:24.570598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:24.570598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ykfuyNVqTm2Q4OvBfnVSdupXgD8eRMNrDoueHSXMPLhzvm4VpYxE6k2e+RaBmi5J1hxShZHbL853A2M23gfxDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:24.571183Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.04079","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4a013f0b92dec97a390ddcaa3cbf7e41c89f3220f03ae1a6d053284e19883b9","sha256:02b772bf7080b28ffa6c3e7ccde827acc5b044422a579cc321b32e150c0ea5c9"],"state_sha256":"fa0a5484c4ad3cf0f9ea2e123d9bd7a2d3f6586b3d2e3e5ea1309ee35eaa6177"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QO31Y8KUp4F1DTaT0Z0KI7XN0xyPZ7EaqCx/Q3+SVdKzzxx27KsBW6EA4eXj5MCrQOEm2aeeaQfjgmFUOOKIBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T14:33:31.907921Z","bundle_sha256":"06f245283968aec3a4cc8307ea12f6ac650cc58c0a568f2a0e1afaf6bea85d8c"}}