91 research outputs found
The word problem for some uncountable groups given by countable words
We investigate the fundamental group of Griffiths' space, and the first
singular homology group of this space and of the Hawaiian Earring by using
(countable) reduced tame words. We prove that two such words represent the same
element in the corresponding group if and only if they can be carried to the
same tame word by a finite number of word transformations from a given list.
This enables us to construct elements with special properties in these groups.
By applying this method we prove that the two homology groups contain
uncountably many different elements that can be represented by infinite
concatenations of countably many commutators of loops. As another application
we give a short proof that these homology groups contain the direct sum of
2^{\aleph_0} copies of \mathbb{Q}. Finally, we show that the fundamental group
of Griffith's space contains \mathbb{Q}.Comment: 24 pages, 7 figure
From local to global conjugacy of subgroups of relatively hyperbolic groups
Suppose that a finitely generated group is hyperbolic relative to a
collection of subgroups . Let be
subgroups of such that is relatively quasiconvex with respect to
and is not parabolic. Suppose that is elementwise
conjugate into . Then there exists a finite index subgroup of which
is conjugate into . The minimal length of the conjugator can be estimated.
In the case where is a limit group, it is sufficient to assume only that
is a finitely generated and is an arbitrary subgroup of .Comment: 14 pages, 1 Figure. The proof in this version is shorte
Orbit decidability, applications and variations
We present the notion of orbit decidability into a more general framework,
exploring interesting generalizations and variations of this algorithmic
problem. A recent theorem by Bogopolski-Martino-Ventura gave a renovated
protagonism to this notion and motivated several interesting algebraic
applications
Orbit decidability and the conjugacy problem for some extensions of groups
Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, weprove that G has solvable conjugacy problem if and only if the corresponding action subgroupA 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable,among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2) is given
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