Random equations in free groups

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

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

In this paper we study the asymptotic probability that a random equation in a finitely generated free group F is solvable in F. For one-variable equations this probability is zero, but for split equations, i.e., equations of the form ν(x1,....,xk) = g, g ∈ F, the probability is strictly between zero and one if k ≥ rank(F) ≥ 2. As a consequence the endomorphism problem in F has intermediate asymptotic density, and we obtain the first natural algebraic examples of subsets of intermediate density in free groups of rank larger than two.

Original languageEnglish
Pages (from-to)257-284
Number of pages28
JournalGroups, Complexity, Cryptology
Volume3
Issue number2
DOIs
StatePublished - Dec 2011

Keywords

  • Asymptotic density
  • Free abelian groups
  • Free groups
  • Random equations

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