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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 v ( x 1 , . . . , x k ) = 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.
Groups - Complexity - Cryptology – de Gruyter
Published: Dec 1, 2011
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