600 research outputs found
Orbit decidability and the conjugacy problem for some extensions of groups
Given a short exact sequence of groups with certain conditions, , we prove that has solvable conjugacy problem if and only if
the corresponding action subgroup is orbit decidable. From
this, we deduce that the conjugacy problem is solvable, among others, for all
groups of the form , , , and with virtually solvable action
group . Also, we give an easy way of constructing
groups of the form and 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 is given
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
Conjugacy classes of solutions to equations and inequations over hyperbolic groups
We study conjugacy classes of solutions to systems of equations and
inequations over torsion-free hyperbolic groups, and describe an algorithm to
recognize whether or not there are finitely many conjugacy classes of solutions
to such a system. The class of immutable subgroups of hyperbolic groups is
introduced, which is fundamental to the study of equations in this context. We
apply our results to enumerate the immutable subgroups of a torsion-free
hyperbolic group.Comment: 28 pages; referee's comments incorporated; to appear in the Journal
of Topolog
The isomorphism problem for all hyperbolic groups
We give a solution to Dehn's isomorphism problem for the class of all
hyperbolic groups, possibly with torsion. We also prove a relative version for
groups with peripheral structures. As a corollary, we give a uniform solution
to Whitehead's problem asking whether two tuples of elements of a hyperbolic
group are in the same orbit under the action of \Aut(G). We also get an
algorithm computing a generating set of the group of automorphisms of a
hyperbolic group preserving a peripheral structure.Comment: 71 pages, 4 figure
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