774 research outputs found
On the finite presentation of subdirect products and the nature of residually free groups
We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion
for the finite presentability of subdirect products of groups: if
are finitely presented and
projects to a subgroup of finite index in
each , then is finitely presentable, indeed there
is an algorithm that will construct a finite presentation for .
We use the VSP criterion to characterise the finitely presented residually
free groups. We prove that the class of such groups is recursively enumerable.
We describe an algorithm that, given a finite presentation of a residually free
group, constructs a canonical embedding into a direct product of finitely many
limit groups. We solve the (multiple) conjugacy problem and membership problem
for finitely presentable subgroups of residually free groups. We also prove
that there is an algorithm that, given a finite generating set for such a
subgroup, will construct a finite presentation.
New families of subdirect products of free groups are constructed, including
the first examples of finitely presented subgroups that are neither
nor of Stallings-Bieri typeComment: 44 pages. To appear in American Journal of Mathematics. This is a
substantial rewrite of our previous Arxiv article 0809.3704, taking into
account subsequent developments, advice of colleagues and referee's comment
On conjugacy separability of fibre products
In this paper we study conjugacy separability of subdirect products of two
free (or hyperbolic) groups. We establish necessary and sufficient criteria and
apply them to fibre products to produce a finitely presented group in
which all finite index subgroups are conjugacy separable, but which has an
index overgroup that is not conjugacy separable. Conversely, we construct a
finitely presented group which has a non-conjugacy separable subgroup of
index such that every finite index normal overgroup of is conjugacy
separable. The normality of the overgroup is essential in the last example, as
such a group will always posses an index overgroup that is not
conjugacy separable.
Finally, we characterize -conjugacy separable subdirect products of two
free groups, where is a prime. We show that fibre products provide a
natural correspondence between residually finite -groups and -conjugacy
separable subdirect products of two non-abelian free groups. As a consequence,
we deduce that the open question about the existence of an infinite finitely
presented residually finite -group is equivalent to the question about the
existence of a finitely generated -conjugacy separable full subdirect
product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio
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
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