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Generically such a group is a small cancellation group and it is clear that $(a_1,...,a_k)$ and $(b_1,...,b_k)$ are generating $n$-tuples for $G$. We prove that for generic choices of $u_1,..., u_k$ and $v_1,..., v_k$ the \"once-stabilized\" tuples $(a_1,..., a_k,1)$ and $(b_1,...,b_k,1)$ are not Nielsen equivalent in $G$. This provides a counter-example for a Wiego"},"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":"1011.5862","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-11-26T19:23:44Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"5859f3bea2f94a6ebba4defc64395ca9739e387cfa00ad0d59e2077f7a2afb03","abstract_canon_sha256":"7d5016e950c240226e891cc10763a5128b7486e90d200a16b4c416ae6b11aaf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:23:17.435987Z","signature_b64":"v3KpXNxdTIGlq1SBVt7Hu2cUoM2dHdnJpe9qG77WdufYmECThPKrjmgSBwEFXiQF/y8YfBhEPuu5wn5/FnzzDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4be6efb413bd15dea045c9703103aac993e88fb7b1ae55571932d3a26e4e879e","last_reissued_at":"2026-05-18T02:23:17.435331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:23:17.435331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nielsen equivalence in small cancellation groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Ilya Kapovich, Richard Weidmann","submitted_at":"2010-11-26T19:23:44Z","abstract_excerpt":"Let $G$ be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\\,| a_i=u_i(\\bar b), b_i=v_i(\\bar a) \\hbox{for} 1\\le i\\le k>,] where $k\\ge 2$ and where the $u_i\\in F(b_1,..., b_k)$ and $w_i\\in F(a_1,..., a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,a_k)$ and $(b_1,...,b_k)$ are generating $n$-tuples for $G$. We prove that for generic choices of $u_1,..., u_k$ and $v_1,..., v_k$ the \"once-stabilized\" tuples $(a_1,..., a_k,1)$ and $(b_1,...,b_k,1)$ are not Nielsen equivalent in $G$. 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