{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:3YZ44Y3HQ5VCDMVGKRMPHGYZAP","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":"418691aafea70de159b8a61ac9531d9e688f3dbdf33ea5182629052ac171b9c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-07-16T23:26:30Z","title_canon_sha256":"77e0c3686bb4fb492cdad45338b51c63929973ac869e198e14f8faf212260e54"},"schema_version":"1.0","source":{"id":"2507.12676","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.12676","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"2507.12676v1","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.12676","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"3YZ44Y3HQ5VC","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_16","alias_value":"3YZ44Y3HQ5VCDMVG","created_at":"2026-07-05T11:38:37Z"},{"alias_kind":"pith_short_8","alias_value":"3YZ44Y3H","created_at":"2026-07-05T11:38:37Z"}],"graph_snapshots":[{"event_id":"sha256:edd0ebcd889d7bd90d78a71156132434102b0b886ed1a9f9cdd49e9d2edcad46","target":"graph","created_at":"2026-07-05T11:38:37Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.12676/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Using dynamical techniques we show that there are no isolated elements on the space of left-preorders on a free product of two groups. As a consequence, when the groups are finitely generated, this space is either empty or a Cantor set. For any subgroup of a free product having finite Kurosh rank, we develop similar results for the subspace consisting on the set of left-preorders relative to that subgroup. We provide a generating set for a certain intersection of subgroups of a free product.","authors_text":"Iv\\'an Ch\\'ercoles Cuesta","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-07-16T23:26:30Z","title":"Spaces of left-preorders on free products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12676","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":"sha256:9207156b5cdbe52a9f7c4474e1626956a4ce63fc51bf4d27e33853035c0052d5","target":"record","created_at":"2026-07-05T11:38:37Z","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":"418691aafea70de159b8a61ac9531d9e688f3dbdf33ea5182629052ac171b9c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-07-16T23:26:30Z","title_canon_sha256":"77e0c3686bb4fb492cdad45338b51c63929973ac869e198e14f8faf212260e54"},"schema_version":"1.0","source":{"id":"2507.12676","kind":"arxiv","version":1}},"canonical_sha256":"de33ce6367876a21b2a65458f39b1903f2b75c6b2155a53eab895f7d0e1f9ca2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de33ce6367876a21b2a65458f39b1903f2b75c6b2155a53eab895f7d0e1f9ca2","first_computed_at":"2026-07-05T11:38:37.101613Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:38:37.101613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aZjecRJerbVvKbwpZ3Yk6fgy6XIPuPSXU1ifWsX2aPVqAaVzCvbFt4+lFPfrpeQthTArk3QEXRrpJkexyrP5AA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:38:37.102073Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.12676","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9207156b5cdbe52a9f7c4474e1626956a4ce63fc51bf4d27e33853035c0052d5","sha256:edd0ebcd889d7bd90d78a71156132434102b0b886ed1a9f9cdd49e9d2edcad46"],"state_sha256":"ce8d725d4f59e41b1d4c24c1805172e10ce5b60f1018a9335a68aee2455b8d5b"}