Random equations in nilpotent groups

Robert H. Gilman, Alexei Myasnikov, Vitali Roman'kov

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. Let Sat(G, k) be the set of all satisfiable equations over G in k variables. For a free abelian group A m of rank m we show that the asymptotic density ρ(Sat(A m, k)) of the set Sat(A m, k) (in the whole space of all equations in k variables over G) is equal to 0 for k=1, and is equal to ζ(k+m)/ζ(k) for k≥2. More generally, if G is a finitely generated nilpotent infinite group, then again the asymptotic density of the set Sat(G, 1) is 0. For k≥2 we give robust estimates for the upper and lower asymptotic densities of the set Sat(G, k). Namely, we prove that these densities lie in the interval from 1t(G)ζ(k+h(G))ζ(k) to ζ(k+m)ζ(k), where h(G) is the Hirsch length of G, t(G) is the order of lower central torsion of G, and m is the torsion-free rank (Hirsch length) of the abelianization of G. This allows one to describe the asymptotic behavior of the set Sat(G, k) for different parameters k, m, h(G), t(G).

Original languageEnglish
Pages (from-to)192-214
Number of pages23
JournalJournal of Algebra
Volume352
Issue number1
DOIs
StatePublished - 15 Feb 2012

Keywords

  • Asymptotic density
  • Equations
  • Free abelian groups
  • Free groups
  • Free nilpotent groups

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