TY - JOUR
T1 - Random equations in free groups
AU - Gilman, Robert H.
AU - Myasnikov, Alexei
AU - Vitaliǐ, Roman'kov
PY - 2011/12
Y1 - 2011/12
N2 - 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.
AB - 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.
KW - Asymptotic density
KW - Free abelian groups
KW - Free groups
KW - Random equations
UR - http://www.scopus.com/inward/record.url?scp=84855243375&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84855243375&partnerID=8YFLogxK
U2 - 10.1515/GCC.2011.010
DO - 10.1515/GCC.2011.010
M3 - Article
AN - SCOPUS:84855243375
SN - 1867-1144
VL - 3
SP - 257
EP - 284
JO - Groups, Complexity, Cryptology
JF - Groups, Complexity, Cryptology
IS - 2
ER -