125 research outputs found
The structure of one-relator relative presentations and their centres
Suppose that G is a nontrivial torsion-free group and w is a word in the
alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from
w by erasing all letters belonging to G is not a proper power in the free group
F(x_1,...,x_n). We show how to reduce the study of the relative presentation
\^G= to the case n=1. It turns out that an
"n-variable" group \^G can be constructed from similar "one-variable" groups
using an explicit construction similar to wreath product. As an illustration,
we prove that, for n>1, the centre of \^G is always trivial. For n=1, the
centre of \^G is also almost always trivial; there are several exceptions, and
all of them are known.Comment: 15 pages. A Russian version of this paper is at
http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm . V4:
the intoduction is rewritten; Section 1 is extended; a short introduction to
Secton 5 is added; some misprints are corrected and some cosmetic
improvements are mad
Free subgroups of one-relator relative presentations
Suppose that G is a nontrivial torsion-free group and w is a word over the
alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the
group \~G= always contains a nonabelian free subgroup.
For n=1 the question about the existence of nonabelian free subgroups in \~G is
answered completely in the unimodular case (i.e., when the exponent sum of x_1
in w is one). Some generalisations of these results are discussed.Comment: V3: A small correction in the last phrase of the proof of Theorem 1.
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