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The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property $(T)$ for $\\operatorname{Aut}F_n$, $n\\geq 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\\ge \\max\\{2,\\operatorname{rank}(G)\\}$ when $G$ is indicable, and for $n$ big enough when $G$ is elementary amenab"},"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":"1309.0271","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-09-01T22:10:31Z","cross_cats_sorted":[],"title_canon_sha256":"ba189b5409f279b32d8798b15ad68c2d4a477a39331877a2e47384782d24227c","abstract_canon_sha256":"e404481e6a324e08887d05110227a94d68684f7bafaabb895a4cb8b6ffae348b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:14.038840Z","signature_b64":"f5igUXK7pOMz43YlZIeJR3FHKB0BjUjbQCjkhBJXKZ0Bj4GDjZOsXZAFOH7jHzqdZEjxedVbfa7BiQOORkHbAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"596268740a6ad9167ccf7d9aaad30584efd5343cb2effc691d4eca002e60f443","last_reissued_at":"2026-05-18T01:21:14.038100Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:14.038100Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On transitivity and (non)amenability of Aut(F_n) actions on group presentations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aglaia Myropolska, Tatiana Nagnibeda","submitted_at":"2013-09-01T22:10:31Z","abstract_excerpt":"For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\\geq \\operatorname{rank}(G)$, describes the action of the group $\\operatorname{Aut}F_n$ of automorphisms of the free group $F_n$ on generating $n$-tuples of G by elementary Nielsen moves. 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