{"paper":{"title":"Random nilpotent groups, polycyclic presentations, and Diophantine problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov","submitted_at":"2016-12-08T14:11:44Z","abstract_excerpt":"We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\\tau_2$-groups). 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02651","kind":"arxiv","version":1},"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"}