{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:XN6UW5U75VWGPNTGVWBTFPAVNY","short_pith_number":"pith:XN6UW5U7","schema_version":"1.0","canonical_sha256":"bb7d4b769fed6c67b666ad8332bc156e2aff393cd13cccaa18eab96280f22068","source":{"kind":"arxiv","id":"1702.01662","version":3},"attestation_state":"computed","paper":{"title":"Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Michael Brandenbursky, Micha{\\l} Marcinkowski","submitted_at":"2017-02-06T15:42:22Z","abstract_excerpt":"Let $F_n$ be the free group on $n$ generators and $\\Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $\\Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem pose"},"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":"1702.01662","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-02-06T15:42:22Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"8df1089ea3bc962f11087da0ff118c8cd952101f3e8559d6d52823962934ae2a","abstract_canon_sha256":"a2ad7a7f9934a10da1f7fdbf3fef4d220e531dead769acfd2fa49c3d1577d58a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:06.372996Z","signature_b64":"U7hoW0ewsMRvpixq7oTZeEo/Jh409h8HukJyHfVt/c4BwKC5UV2rmlQBPCv6nMHoJKDF92ZSJjAzNHM/27pACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb7d4b769fed6c67b666ad8332bc156e2aff393cd13cccaa18eab96280f22068","last_reissued_at":"2026-05-18T00:06:06.372455Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:06.372455Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Michael Brandenbursky, Micha{\\l} Marcinkowski","submitted_at":"2017-02-06T15:42:22Z","abstract_excerpt":"Let $F_n$ be the free group on $n$ generators and $\\Gamma_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $\\Gamma_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem pose"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01662","kind":"arxiv","version":3},"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":"1702.01662","created_at":"2026-05-18T00:06:06.372538+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.01662v3","created_at":"2026-05-18T00:06:06.372538+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.01662","created_at":"2026-05-18T00:06:06.372538+00:00"},{"alias_kind":"pith_short_12","alias_value":"XN6UW5U75VWG","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_16","alias_value":"XN6UW5U75VWGPNTG","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_8","alias_value":"XN6UW5U7","created_at":"2026-05-18T12:31:56.362134+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/XN6UW5U75VWGPNTGVWBTFPAVNY","json":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY.json","graph_json":"https://pith.science/api/pith-number/XN6UW5U75VWGPNTGVWBTFPAVNY/graph.json","events_json":"https://pith.science/api/pith-number/XN6UW5U75VWGPNTGVWBTFPAVNY/events.json","paper":"https://pith.science/paper/XN6UW5U7"},"agent_actions":{"view_html":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY","download_json":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY.json","view_paper":"https://pith.science/paper/XN6UW5U7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.01662&json=true","fetch_graph":"https://pith.science/api/pith-number/XN6UW5U75VWGPNTGVWBTFPAVNY/graph.json","fetch_events":"https://pith.science/api/pith-number/XN6UW5U75VWGPNTGVWBTFPAVNY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY/action/storage_attestation","attest_author":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY/action/author_attestation","sign_citation":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY/action/citation_signature","submit_replication":"https://pith.science/pith/XN6UW5U75VWGPNTGVWBTFPAVNY/action/replication_record"}},"created_at":"2026-05-18T00:06:06.372538+00:00","updated_at":"2026-05-18T00:06:06.372538+00:00"}