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Our proof relies on the classification of finite simple groups. For $A,B$ that are finitely presented we show that $$ \\lim_{n \\to \\infty} \\frac{\\log |\\mathrm{Torsion}(U_n^{\\mathrm{ab}})|}{[G : U_n]} = 0. $$"},"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":"1609.08900","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-09-28T13:14:34Z","cross_cats_sorted":[],"title_canon_sha256":"46feb16c9ac4f847eed0c25711d6751f33e2e2f51fe0f2548dcb794b9ad8a0db","abstract_canon_sha256":"a6c169e51435dcaeb30cf7d214105527f95959bcf94e4c5b4b5d66622a501aa7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:38.833426Z","signature_b64":"u4cliLkTfJSd/rcn9YRPJ4EIFCIHNZVDH3kYlK8/jFMfuJa8WElMYlhxpTfdmRC8p3PIQrFvTmSKPIUPQbg1Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e08afb9b493a407fb872bcb28092036fe68153a16a3706b6fab89b5a2e907a6f","last_reissued_at":"2026-05-18T00:44:38.832961Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:38.832961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gradients of sequences of subgroups in a direct product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Mark Shusterman, Nikolay Nikolov, Zvi Shemtov","submitted_at":"2016-09-28T13:14:34Z","abstract_excerpt":"For a sequence $\\{U_n\\}_{n = 1}^\\infty$ of finite index subgroups of a direct product $G = A \\times B$ of finitely generated groups, we show that $$\\lim_{n \\to \\infty} \\frac{\\min\\{|X| : \\langle X \\rangle = U_n\\}}{[G : U_n]} = 0$$ once $[A : A \\cap U_n], [B : B \\cap U_n] \\to \\infty$ as $n \\to \\infty$. Our proof relies on the classification of finite simple groups. For $A,B$ that are finitely presented we show that $$ \\lim_{n \\to \\infty} \\frac{\\log |\\mathrm{Torsion}(U_n^{\\mathrm{ab}})|}{[G : U_n]} = 0. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08900","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.08900","created_at":"2026-05-18T00:44:38.833029+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.08900v2","created_at":"2026-05-18T00:44:38.833029+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.08900","created_at":"2026-05-18T00:44:38.833029+00:00"},{"alias_kind":"pith_short_12","alias_value":"4CFPXG2JHJAH","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"4CFPXG2JHJAH7ODS","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"4CFPXG2J","created_at":"2026-05-18T12:29:58.707656+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/4CFPXG2JHJAH7ODSXSZIBEQDN7","json":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7.json","graph_json":"https://pith.science/api/pith-number/4CFPXG2JHJAH7ODSXSZIBEQDN7/graph.json","events_json":"https://pith.science/api/pith-number/4CFPXG2JHJAH7ODSXSZIBEQDN7/events.json","paper":"https://pith.science/paper/4CFPXG2J"},"agent_actions":{"view_html":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7","download_json":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7.json","view_paper":"https://pith.science/paper/4CFPXG2J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.08900&json=true","fetch_graph":"https://pith.science/api/pith-number/4CFPXG2JHJAH7ODSXSZIBEQDN7/graph.json","fetch_events":"https://pith.science/api/pith-number/4CFPXG2JHJAH7ODSXSZIBEQDN7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7/action/storage_attestation","attest_author":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7/action/author_attestation","sign_citation":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7/action/citation_signature","submit_replication":"https://pith.science/pith/4CFPXG2JHJAH7ODSXSZIBEQDN7/action/replication_record"}},"created_at":"2026-05-18T00:44:38.833029+00:00","updated_at":"2026-05-18T00:44:38.833029+00:00"}