Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / 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 / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2017: 0.32

See all formats and pricing
More options …

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


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.

Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

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

Comments (0)

Please log in or register to comment.
Log in