84 research outputs found
On several problems about automorphisms of the free group of rank two
Let be a free group of rank . In this paper we discuss three
algorithmic problems related to automorphisms of .
A word of is called positive if does not have negative
exponents. A word in is called potentially positive if is
positive for some automorphism of . We prove that there is an
algorithm to decide whether or not a given word in is potentially
positive, which gives an affirmative solution to problem F34a in [1] for the
case of .
Two elements and in are said to be boundedly translation
equivalent if the ratio of the cyclic lengths of and is
bounded away from 0 and from for every automorphism of .
We provide an algorithm to determine whether or not two given elements of
are boundedly translation equivalent, thus answering question F38c in the
online version of [1] for the case of .
We further prove that there exists an algorithm to decide whether or not a
given finitely generated subgroup of is the fixed point group of some
automorphism of , which settles problem F1b in [1] in the affirmative for
the case of .Comment: 30 page
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.Comment: 36 page
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= \phi(u) for some \phi \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.Comment: 35 pages, revised versio
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