TY - JOUR
T1 - Random equations in nilpotent groups
AU - Gilman, Robert H.
AU - Myasnikov, Alexei
AU - Roman'kov, Vitali
PY - 2012/2/15
Y1 - 2012/2/15
N2 - 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).
AB - 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).
KW - Asymptotic density
KW - Equations
KW - Free abelian groups
KW - Free groups
KW - Free nilpotent groups
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U2 - 10.1016/j.jalgebra.2011.11.007
DO - 10.1016/j.jalgebra.2011.11.007
M3 - Article
AN - SCOPUS:84855210617
SN - 0021-8693
VL - 352
SP - 192
EP - 214
JO - Journal of Algebra
JF - Journal of Algebra
IS - 1
ER -