18 research outputs found
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.
4 page
Economical adjunction of square roots to groups
How large must an overgroup of a given group be in order to contain a square
root of any element of the initial group? We give an almost exact answer to
this question (the obtained estimate is at most twice worse than the best
possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at
http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2:
minor correction
Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator
A group obtained from a nontrivial group by adding one generator and one
relator which is a proper power of a word in which the exponent-sum of the
additional generator is one contains the free square of the initial group and
almost always (with one obvious exception) contains a non-abelian free
subgroup. If the initial group is involution-free or the relator is at least
third power, then the obtained group is SQ-universal and relatively hyperbolic
with respect to the initial group.Comment: 11 pages. A Russian version of this paper is at
http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V3:
revised following referee's comment
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
Equations over solvable groups
Not any nonsingular equation over a metabelian group has solution in a larger
metabelian group. However, any nonsingular equation over a solvable group with
a subnormal series with abelian torsion-free quotients has a solution in a
larger group with a similar subnormal series of the same length (and an
analogous fact is valid for systems of equations).Comment: 7 pages. A Russian version of this paper is at
http://halgebra.math.msu.su/staff/klyachko/papers.htm . V3: misprints
correcte
Yet another Freiheitssatz: Mating finite groups with locally indicable ones
The main result includes as special cases on the one hand, the
Gerstenhaber--Rothaus theorem (1962) and its generalisation due to Nitsche and
Thom (2022) and, on the other hand, the Brodskii--Howie--Short theorem
(1980--1984) generalising Magnus's Freiheitssatz (1930).Comment: 5 pages. A Russian version of this paper is at
http://halgebra.math.msu.su/staff/klyachko/papers.ht