Some Results on Free Groups in Combinatorial Group Theory

Let Fm be a free group of a finite rank m ≥ 2. We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup ⟨X1,..., Xn⟩ of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Lee, Donghi

Contributors dc:contributor

• Sergei v. Ivanov

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3017139

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86777