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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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CiteScore 2017: 0.32

SCImago Journal Rank (SJR) 2017: 0.208
Source Normalized Impact per Paper (SNIP) 2017: 0.322

Mathematical Citation Quotient (MCQ) 2017: 0.32

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1869-6104
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Random equations in free groups

Robert H. Gilman / Alexei Myasnikov / Roman'kov Vitali
Published Online: 2011-11-25 | DOI: https://doi.org/10.1515/gcc.2011.010

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, . . . , xk ) = g, gF, 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.

Keywords.: Free abelian groups; free groups; random equations; asymptotic density

About the article

Received: 2011-05-12

Published Online: 2011-11-25

Published in Print: 2011-12-01


Citation Information: Groups – Complexity – Cryptology, Volume 3, Issue 2, Pages 257–284, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc.2011.010.

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[1]
Robert H. Gilman, Alexei Myasnikov, and Vitaliĭ Romanʼkov
Journal of Algebra, 2012, Volume 352, Number 1, Page 192
[2]
A. V. Men’shov
Siberian Mathematical Journal, 2014, Volume 55, Number 3, Page 440

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