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To do so, we show that these are precisely the groups that admit a presentation of the form $ \\label{tau2pres_0}\\langle A, C \\mid [a_i, a_j]= \\prod_t {\\scriptstyle c_t^{\\scriptscriptstyle \\lambda_{t,i,j}}} \\ (i< j), \\ [A,C]=[C,C]=1\\rangle,$ where $A=\\{a_1, \\dots, a_n\\}$, and $C=\\{c_1, \\dots, c_m\\}$. Hence, one may select a random $\\tau_2$-group $G$ by fixing $A$ and $C$, and then randomly choosing exponents $\\lambda_{t,i,j}$ with $|\\lambda_{t,i,j}|\\leq \\ell$, for some $\\ell$.\n  We prove th","authors_text":"Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-08T14:11:44Z","title":"Random nilpotent groups, polycyclic presentations, and Diophantine problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02651","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:142e8bfdac33b1632a8727a96f22b1729259f1192bdc2abda777e91f36d61bf5","target":"record","created_at":"2026-05-18T00:55:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"70153bc1a9653cf0a6afb8963f656483480c919fc049a07474a1a04615d7cd62","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-08T14:11:44Z","title_canon_sha256":"396610571a334bd45f02400dfc676c1d6a2a22b6692c7792b544888c2dd05179"},"schema_version":"1.0","source":{"id":"1612.02651","kind":"arxiv","version":1}},"canonical_sha256":"1ed6d955ea02346de309eb710f1737b41498949a7b015246b44163b357316cb5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ed6d955ea02346de309eb710f1737b41498949a7b015246b44163b357316cb5","first_computed_at":"2026-05-18T00:55:33.169629Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:33.169629Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NVZ9/YI6U1FBYVuYh5NcqIGIBlSmrU51HnjElJtOSE8o5cRSN+Q8xoB2lrBsCFxuWjtzHWQmuRtLSVE6P4u1Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:33.170155Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.02651","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:142e8bfdac33b1632a8727a96f22b1729259f1192bdc2abda777e91f36d61bf5","sha256:bccfcca8604d653a8487d5c28ef59f1f8e6ce4bf51230386aaaf417929f32c3e"],"state_sha256":"ee292ffffe03f916a42fe8767db63643c218b09e610f48441aed1633cfe369fc"}