{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2NGWJVOUU6XZSAPM6QLFYSLMYD","short_pith_number":"pith:2NGWJVOU","schema_version":"1.0","canonical_sha256":"d34d64d5d4a7af9901ecf4165c496cc0ce4f890aa64eefde770fc987e750f779","source":{"kind":"arxiv","id":"1606.07728","version":1},"attestation_state":"computed","paper":{"title":"Groups of piecewise isometric permutations of lattice points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Heike Sach, Robert Bieri","submitted_at":"2016-06-24T15:51:43Z","abstract_excerpt":"Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in M, we consider the orbit Gp in M and define a notion of \"G-polyhedral pieces\" S in Gp. The objects of our interest are the groups pi(S) of all piecewise G-isometric permutations on S.\n  In this paper we merely present the two most basic examples, and these play rather different roles: The case when G = PSL(2,Z) acting on the hyperbolic plane reveals that the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.07728","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-06-24T15:51:43Z","cross_cats_sorted":[],"title_canon_sha256":"57861011286b0fbfeb2d5912061602e95268d33605497a2c4ce1bf7cac7e6dca","abstract_canon_sha256":"28ae241760a062698f2bc54324be211bfa84819ce2ce1e5aafc78c9f33204637"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:55.816823Z","signature_b64":"+gEOcdfoGnE2OHmToQjDXTfh8yYpVncrow/HvJvV347MvEo3HTObzlAg+aVtrLaq5UcNojtWpIK5Isam0GViCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d34d64d5d4a7af9901ecf4165c496cc0ce4f890aa64eefde770fc987e750f779","last_reissued_at":"2026-05-18T01:11:55.816492Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:55.816492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Groups of piecewise isometric permutations of lattice points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Heike Sach, Robert Bieri","submitted_at":"2016-06-24T15:51:43Z","abstract_excerpt":"Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in M, we consider the orbit Gp in M and define a notion of \"G-polyhedral pieces\" S in Gp. The objects of our interest are the groups pi(S) of all piecewise G-isometric permutations on S.\n  In this paper we merely present the two most basic examples, and these play rather different roles: The case when G = PSL(2,Z) acting on the hyperbolic plane reveals that the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07728","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.07728","created_at":"2026-05-18T01:11:55.816546+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.07728v1","created_at":"2026-05-18T01:11:55.816546+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07728","created_at":"2026-05-18T01:11:55.816546+00:00"},{"alias_kind":"pith_short_12","alias_value":"2NGWJVOUU6XZ","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2NGWJVOUU6XZSAPM","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2NGWJVOU","created_at":"2026-05-18T12:29:55.572404+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD","json":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD.json","graph_json":"https://pith.science/api/pith-number/2NGWJVOUU6XZSAPM6QLFYSLMYD/graph.json","events_json":"https://pith.science/api/pith-number/2NGWJVOUU6XZSAPM6QLFYSLMYD/events.json","paper":"https://pith.science/paper/2NGWJVOU"},"agent_actions":{"view_html":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD","download_json":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD.json","view_paper":"https://pith.science/paper/2NGWJVOU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.07728&json=true","fetch_graph":"https://pith.science/api/pith-number/2NGWJVOUU6XZSAPM6QLFYSLMYD/graph.json","fetch_events":"https://pith.science/api/pith-number/2NGWJVOUU6XZSAPM6QLFYSLMYD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD/action/storage_attestation","attest_author":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD/action/author_attestation","sign_citation":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD/action/citation_signature","submit_replication":"https://pith.science/pith/2NGWJVOUU6XZSAPM6QLFYSLMYD/action/replication_record"}},"created_at":"2026-05-18T01:11:55.816546+00:00","updated_at":"2026-05-18T01:11:55.816546+00:00"}