arXiv subjects

Donghi Lee

Publications and source records attributed to Donghi Lee.

26 records · Page 2Linked to original sources

arXiv2011

Simple loops on 2-bridge spheres in 2-bridge link complements

The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane's identity.

math.GT↗

arXiv2008

On several problems about automorphisms of the free group of rank two

Let $F_n$ be a free group of rank $n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$. A word $u$ of $F_n$ is called positive if $u$ does not have negative exponents. A word $u$ in $F_n$ is called potentially positive if $ϕ(u)$ is positive for some automorphism $ϕ$ of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of $F_2$. Two elements $u$ and $v$ in $F_n$ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of $ϕ(u)$ and $ϕ(v)$ is bounded away from 0 and from $\infty$ for every automorphism $ϕ$ of $F_n$. We provide an algorithm to determine whether or not two given elements of $F_2$ are boundedly translation equivalent, thus answering question F38c in the online version of [1] for the case of $F_2$. We further prove that there exists an algorithm to decide whether or not a given finitely generated subgroup of $F_2$ is the fixed point group of some automorphism of $F_2$, which settles problem F1b in [1] in the affirmative for the case of $F_2$.

math.GR↗

arXiv2006

An algorithm that decides translation equivalence in a free group of rank two

Let F_2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for given two elements u, v of F_2, u and v are translation equivalent in F_2, that is, whether or not u and v have the property that the cyclic length of phi(u) equals the cyclic length of phi(v) for every automorphism phi of F_2. This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [1] for the case of F_2.

math.GR↗

arXiv2006

Translation equivalent elements in free groups

Let F_n be a free group of rank n>1. Two elements g, h in F_n are said to be translation equivalent in F_n if the cyclic length of ϕ(g) equals the cyclic length of ϕ(h) for every automorphism ϕof F_n. Let F(a, b) be the free group generated by {a, b} and let w(a,b) be an arbitrary word in F(a,b). We prove that w(g, h) and w(h, g) are translation equivalent in F_n whenever g, h \in F_n are translation equivalent in F_n, which hereby gives an affirmative solution to problem F38b in the online version (http://www.grouptheory.info) of [1].

math.GR↗

arXiv2006

Counting words of minimum length in an automorphic orbit

Let u be a cyclic word in a free group F_n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v|=|u| and v= ϕ(u) for some ϕ\in \text {Aut}F_n}. In this paper, we prove that N(u) is bounded by a polynomial function with respect to |u| under the hypothesis that if two letters x, y occur in u, then the total number of x and x^{-1} occurring in u is not equal to the total number of y and y^{-1} occurring in u. A complete proof without the hypothesis would yield the polynomial time complexity of Whitehead's algorithm for F_n.

math.GR↗

arXiv2006

A tighter bound for the number of words of minimum length in an automorphic orbit

Let u be a cyclic word in a free group F_n of finite rank n that has the minimum length over all cyclic words in its automorphic orbit, and let N(u) be the cardinality of the set {v: |v|=|u| and v=ϕ(u) for some ϕ\in AutF_n}. In this paper, we prove that N(u) is bounded by a polynomial function of degree 2n-3 with respect to |u| under the hypothesis that if two letters x, y occur in u, then the total number of x and x^{-1} occurring in u is not equal to the total number of y and y^{-1} occurring in u. We also prove that 2n-3 is the sharp bound on the degree of polynomials bounding N(u). As a special case, we deal with N(u) in F_2 under the same hypothesis.

math.GR↗

arXiv2001

On certain C-test words for free groups

Let F_m be a free group of a finite rank m > 1 and X_i, Y_j be elements in F_m. A non-empty word w(x_1,..., x_n) is called a C-test word in n letters for F_m if, whenever w(X_1,..., X_n)=w(Y_1,..., Y_n) not equal to 1, the two n-tuples (X_1,..., X_n) and (Y_1,..., Y_n) are conjugate in F_m. In this paper we construct, for each n > 1, a C-test word v_n(x_1,..., x_n) with the additional property that v_n(X_1,..., X_n)=1 if and only if the subgroup of F_m generated by X_1,..., X_n is cyclic. Making use of such words v_m(x_1,..., x_m) and v_{m+1}(x_1,..., x_{m+1}), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u_1, u_2 in F_m such that every endomorphism of F_m with non-cyclic image is completely determined by its values on u_1, u_2.

math.GR↗

arXiv2001

Primitivity preserving endomorphisms of free groups

An endomorphism of a free group is called primitivity preserving if it takes every primitive element to another primitive. In this paper we prove that every primitivity preserving endomorphism of a free group of a finite rank n > 2 is an automorphism.

math.GR↗