Random equations in nilpotent groups

2011/05/11 by Robert Gilman, Gilman, Robert, Alexei Myasnikov +3
Mathematics · #20F70 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20F70

paper · pdf · doi:10.48550/arxiv.1105.2234

25 pages

arxiv created 2011/06/08 · arxiv updated 2011/06/10

Abstract

In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. We show that the set SAT(G,k) of all equations in k > 1 variables over G which are satisfiable in G has an intermediate asymptotic density in the space of all equations in k variables over G. When G is a free abelian group of finite rank, we compute this density precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1 the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting sets in groups are either zero or one. The results of this paper provide new examples of algebraically significant sets of intermediate asymptotic density.

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