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Random equations in free groups

Random equations in free groups 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups - Complexity - Cryptology de Gruyter

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Publisher
de Gruyter
Copyright
Copyright © 2011 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc.2011.010
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Groups - Complexity - Cryptologyde Gruyter

Published: Dec 1, 2011

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