{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:TWWAUN2RPRB7HRFWSSEJ6LXB3M","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":"03f80af36c44bac3f9bc82ad8671a124c2380ae629eb378b1fe1a3dd141e7e22","cross_cats_sorted":["math.DS","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-05-08T11:11:45Z","title_canon_sha256":"6b6f9ebde234fac1b37bf04596e472ae69040e47326e2a50f0587740eb7094d6"},"schema_version":"1.0","source":{"id":"1405.1881","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1881","created_at":"2026-05-18T01:38:54Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1881v2","created_at":"2026-05-18T01:38:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1881","created_at":"2026-05-18T01:38:54Z"},{"alias_kind":"pith_short_12","alias_value":"TWWAUN2RPRB7","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"TWWAUN2RPRB7HRFW","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"TWWAUN2R","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:36337837becbe6b331549e3b4c4aacf803d734e6007066a9d5f84fb66b3ea3f6","target":"graph","created_at":"2026-05-18T01:38:54Z","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 concept of a generic Euclidean triangle $\\tau$ and study the group $G_\\tau$ generated by the reflection across the edges of $\\tau$. In particular, we prove that the subgroup $T_\\tau$ of all translations in $G_\\tau$ is free abelian of infinite rank, while the index 2 subgroup $H_\\tau$ of all orientation preserving transformations in $G_\\tau$ is free metabelian of rank 2, with $T_\\tau$ as the commutator subgroup. As a consequence, the group $G_\\tau$ cannot be finitely presented and we provide explicit minimal infinite presentations of both $H_\\tau$ and $G_\\tau$. This answers in ","authors_text":"Riccardo Piergallini, Stefano Isola","cross_cats":["math.DS","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-05-08T11:11:45Z","title":"On the generic triangle group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1881","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":"sha256:3d94bb0817d2efe39d12df7db1dc7255b3178ecad71ee8468c545fa8ff380849","target":"record","created_at":"2026-05-18T01:38:54Z","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":"03f80af36c44bac3f9bc82ad8671a124c2380ae629eb378b1fe1a3dd141e7e22","cross_cats_sorted":["math.DS","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-05-08T11:11:45Z","title_canon_sha256":"6b6f9ebde234fac1b37bf04596e472ae69040e47326e2a50f0587740eb7094d6"},"schema_version":"1.0","source":{"id":"1405.1881","kind":"arxiv","version":2}},"canonical_sha256":"9dac0a37517c43f3c4b694889f2ee1db33e034ceeeab637cf402530f8283c3bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9dac0a37517c43f3c4b694889f2ee1db33e034ceeeab637cf402530f8283c3bb","first_computed_at":"2026-05-18T01:38:54.286617Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:54.286617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AiYrfNzZxrz0a2dE7JPaZAns73JwH6v91lWNfw38LgPgM/G+GPjIH6DbQgWijkVPGOG+9LB7iv52BHnvrCBGAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:54.287469Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.1881","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d94bb0817d2efe39d12df7db1dc7255b3178ecad71ee8468c545fa8ff380849","sha256:36337837becbe6b331549e3b4c4aacf803d734e6007066a9d5f84fb66b3ea3f6"],"state_sha256":"9b547a4d8b1b8533cfecad45ce6f4010cbec752801e60aff085a97508709ebd4"}