Random equations in nilpotent groups

TL;DR

This paper investigates the probability that random equations in finitely generated nilpotent groups are satisfiable, revealing intermediate asymptotic densities and providing precise calculations for free abelian groups.

Contribution

It introduces the concept of intermediate asymptotic density for satisfiable equations in nilpotent groups and computes exact densities for free abelian groups, offering new insights into algebraic set distributions.

Findings

Set SAT(G,k) has intermediate asymptotic density for k > 1

Exact density computed for free abelian groups

Provides bounds for non-abelian nilpotent groups

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.