{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7EKD4XFMTRX6T5CMDBSR7FCJUS","short_pith_number":"pith:7EKD4XFM","canonical_record":{"source":{"id":"1304.1274","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-04T08:53:25Z","cross_cats_sorted":[],"title_canon_sha256":"f2bbdf061ef21624e6b182b605a1c09c6215c44349a610263fa1f29b910acd8b","abstract_canon_sha256":"dd926a7027a4f5a1516ffba34b61f7838cbf11269310063712ca513b3acc1f77"},"schema_version":"1.0"},"canonical_sha256":"f9143e5cac9c6fe9f44c18651f9449a4a9dc5ba96221a75920bd725daf8cd2f7","source":{"kind":"arxiv","id":"1304.1274","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1274","created_at":"2026-05-18T03:27:00Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1274v2","created_at":"2026-05-18T03:27:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1274","created_at":"2026-05-18T03:27:00Z"},{"alias_kind":"pith_short_12","alias_value":"7EKD4XFMTRX6","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7EKD4XFMTRX6T5CM","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7EKD4XFM","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7EKD4XFMTRX6T5CMDBSR7FCJUS","target":"record","payload":{"canonical_record":{"source":{"id":"1304.1274","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-04T08:53:25Z","cross_cats_sorted":[],"title_canon_sha256":"f2bbdf061ef21624e6b182b605a1c09c6215c44349a610263fa1f29b910acd8b","abstract_canon_sha256":"dd926a7027a4f5a1516ffba34b61f7838cbf11269310063712ca513b3acc1f77"},"schema_version":"1.0"},"canonical_sha256":"f9143e5cac9c6fe9f44c18651f9449a4a9dc5ba96221a75920bd725daf8cd2f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:00.808757Z","signature_b64":"e9d9GfWj7Li99kwqKlSXZcpzNYPfCGgBBMBDWufys+t2yNoP3Y4DpWImo9Y9IBDhx1i0gHTe+wk+R21vY4UODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9143e5cac9c6fe9f44c18651f9449a4a9dc5ba96221a75920bd725daf8cd2f7","last_reissued_at":"2026-05-18T03:27:00.808304Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:00.808304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.1274","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":"2026-05-18T03:27:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TU/5G42fyzQxIzuT4CZTjTBZXpP7ptxPb/A6PbbaEctDO1Wx+BeLDNiQf4wwLyRS9ISAWNt+NrEf58lFOZykDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:58:02.093823Z"},"content_sha256":"27378477ebd71ae0959a6e205f49199904302e16964008fb820349bafe18fe98","schema_version":"1.0","event_id":"sha256:27378477ebd71ae0959a6e205f49199904302e16964008fb820349bafe18fe98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7EKD4XFMTRX6T5CMDBSR7FCJUS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Free subgroups acting properly discontinuously","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Zoran Sunic","submitted_at":"2013-04-04T08:53:25Z","abstract_excerpt":"Given an action of a group G on a topological space X, we establish a necessary and sufficient condition for the existence of a free subgroup F of rank 2 of G acting properly discontinuously on at least one nonempty, open, F-invariant subspace of X. In the case of a discrete topology (group action on a set X), the condition simply detects free subgroups of rank 2 acting freely on some orbit."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1274","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":"2026-05-18T03:27:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8oAfOjKAnVnL3z/Ci9Lo6eDCn/9VNi1/sLPre4KUDfCkmVaFw8gmdYwO7GO8fmY7yDVooxX81CLiuBoO5PBiCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:58:02.094546Z"},"content_sha256":"11ceff01a1655210b096347a74c6484869e75a49cb61feda81bf36f9009b7d35","schema_version":"1.0","event_id":"sha256:11ceff01a1655210b096347a74c6484869e75a49cb61feda81bf36f9009b7d35"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS/bundle.json","state_url":"https://pith.science/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS/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":"2026-05-31T09:58:02Z","links":{"resolver":"https://pith.science/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS","bundle":"https://pith.science/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS/bundle.json","state":"https://pith.science/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7EKD4XFMTRX6T5CMDBSR7FCJUS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7EKD4XFMTRX6T5CMDBSR7FCJUS","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":"dd926a7027a4f5a1516ffba34b61f7838cbf11269310063712ca513b3acc1f77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-04T08:53:25Z","title_canon_sha256":"f2bbdf061ef21624e6b182b605a1c09c6215c44349a610263fa1f29b910acd8b"},"schema_version":"1.0","source":{"id":"1304.1274","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1274","created_at":"2026-05-18T03:27:00Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1274v2","created_at":"2026-05-18T03:27:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1274","created_at":"2026-05-18T03:27:00Z"},{"alias_kind":"pith_short_12","alias_value":"7EKD4XFMTRX6","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7EKD4XFMTRX6T5CM","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7EKD4XFM","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:11ceff01a1655210b096347a74c6484869e75a49cb61feda81bf36f9009b7d35","target":"graph","created_at":"2026-05-18T03:27:00Z","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 an action of a group G on a topological space X, we establish a necessary and sufficient condition for the existence of a free subgroup F of rank 2 of G acting properly discontinuously on at least one nonempty, open, F-invariant subspace of X. In the case of a discrete topology (group action on a set X), the condition simply detects free subgroups of rank 2 acting freely on some orbit.","authors_text":"Zoran Sunic","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-04T08:53:25Z","title":"Free subgroups acting properly discontinuously"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1274","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":"sha256:27378477ebd71ae0959a6e205f49199904302e16964008fb820349bafe18fe98","target":"record","created_at":"2026-05-18T03:27:00Z","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":"dd926a7027a4f5a1516ffba34b61f7838cbf11269310063712ca513b3acc1f77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-04T08:53:25Z","title_canon_sha256":"f2bbdf061ef21624e6b182b605a1c09c6215c44349a610263fa1f29b910acd8b"},"schema_version":"1.0","source":{"id":"1304.1274","kind":"arxiv","version":2}},"canonical_sha256":"f9143e5cac9c6fe9f44c18651f9449a4a9dc5ba96221a75920bd725daf8cd2f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9143e5cac9c6fe9f44c18651f9449a4a9dc5ba96221a75920bd725daf8cd2f7","first_computed_at":"2026-05-18T03:27:00.808304Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:00.808304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e9d9GfWj7Li99kwqKlSXZcpzNYPfCGgBBMBDWufys+t2yNoP3Y4DpWImo9Y9IBDhx1i0gHTe+wk+R21vY4UODw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:00.808757Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.1274","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27378477ebd71ae0959a6e205f49199904302e16964008fb820349bafe18fe98","sha256:11ceff01a1655210b096347a74c6484869e75a49cb61feda81bf36f9009b7d35"],"state_sha256":"7473e873a35a4ddd72ff07253720c451e3fc06e118f1ff356bec9a3feaddd26f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u6XUXgBkzvbcKeNuymMWSBwYv/NrDMkWg4m4Sr9CqRYie1qqxc4qlJy8fE4hdQSooRkCG8PdvFwUBpFv2OflCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T09:58:02.097597Z","bundle_sha256":"0dbca21d617f4a3248bee2d98c3bcdf300179c2c481c4ab5024e1e755762033e"}}