12,539 research outputs found
Strongly-cyclic branched coverings of (1,1)-knots and cyclic presentations of groups
We study the connections among the mapping class group of the twice punctured
torus, the cyclic branched coverings of (1,1)-knots and the cyclic
presentations of groups. We give the necessary and sufficient conditions for
the existence and uniqueness of the n-fold strongly-cyclic branched coverings
of (1,1)-knots, through the elements of the mapping class group. We prove that
every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic
presentation for the fundamental group, arising from a Heegaard splitting of
genus n. Moreover, we give an algorithm to produce the cyclic presentation and
illustrate it in the case of cyclic branched coverings of torus knots of type
(k,hk+1) and (k,hk-1).Comment: 16 pages, 2 figures. to appear in the Mathematical Proceedings of the
Cambridge Philosophical Societ
Decision problems and profinite completions of groups
We consider pairs of finitely presented, residually finite groups
P\hookrightarrow\G for which the induced map of profinite completions \hat
P\to \hat\G is an isomorphism. We prove that there is no algorithm that, given
an arbitrary such pair, can determine whether or not is isomorphic to \G.
We construct pairs for which the conjugacy problem in \G can be solved in
quadratic time but the conjugacy problem in is unsolvable.
Let be the class of super-perfect groups that have a compact
classifying space and no proper subgroups of finite index. We prove that there
does not exist an algorithm that, given a finite presentation of a group \G
and a guarantee that \G\in\mathcal J, can determine whether or not
\G\cong\{1\}.
We construct a finitely presented acyclic group \H and an integer such
that there is no algorithm that can determine which -generator subgroups of
\H are perfect
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
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