1,353 research outputs found
Conjugacy in normal subgroups of hyperbolic groups
Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G.
We establish criteria for N to have solvable conjugacy problem and be conjugacy
separable in terms of the corresponding properties of G/N. We show that the
hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction
is hereditarily conjugacy separable. We then use this construction to produce
first examples of finitely generated and finitely presented conjugacy separable
groups that contain non-(conjugacy separable) subgroups of finite index.Comment: Version 3: 18 pages; corrected a problem with justification of
Corollary 8.
Subgroups of direct products of limit groups over Droms RAAGs
A result of Bridson, Howie, Miller and Short states that if is a subgroup
of type of the direct product of limit groups over
free groups, then is virtually the direct product of limit groups over free
groups. Furthermore, they characterise finitely presented residually free
groups. In this paper these results are generalised to limit groups over Droms
right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with
the property that all of their finitely generated subgroups are again RAAGs. In
addition, we show that the generalised conjugacy problem is solvable for
finitely presented groups that are residually a Droms RAAG and that the
membership problem is decidable for their finitely presented subgroups
Subgroups of direct products of limit groups over Droms RAAGs
A result of Bridson, Howie, Miller and Short states that if S is a subgroup of type FPn(Q) of the direct product of n limit groups over free groups, then S is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG, and that their finitely presentable subgroups are separable
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
The membership problem for 3-manifold groups is solvable
We show that the Membership Problem for finitely generated subgroups of
3-manifold groups is solvable.Comment: 19 page
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
Structure and finiteness properties of subdirect products of groups
We investigate the structure of subdirect products of groups, particularly
their finiteness properties. We pay special attention to the subdirect products
of free groups, surface groups and HNN extensions. We prove that a finitely
presented subdirect product of free and surface groups virtually contains a
term of the lower central series of the direct product or else fails to
intersect one of the direct summands. This leads to a characterization of the
finitely presented subgroups of the direct product of 3 free or surface groups,
and to a solution to the conjugacy problem for arbitrary finitely presented
subgroups of direct products of surface groups. We obtain a formula for the
first homology of a subdirect product of two free groups and use it to show
there is no algorithm to determine the first homology of a finitely generated
subgroup.Comment: 29 pages, no figure
The Geometry of the Conjugacy Problem in Wreath Products and Free Solvable Groups
We describe an effective version of the conjugacy problem and study it for
wreath products and free solvable groups. The problem involves estimating the
length of short conjugators between two elements of the group, a notion which
leads to the definition of the conjugacy length function. We show that for free
solvable groups the conjugacy length function is at most cubic. For wreath
products the behaviour depends on the conjugacy length function of the two
groups involved, as well as subgroup distortion within the quotient group.Comment: 24 pages, 4 figures. This was formed from the splitting of
arXiv:1202.5343, titled "On the Magnus Embedding and the Conjugacy Length
Function of Wreath Products and Free Solvable Groups," into two papers. The
contents of this paper remain largely unchange
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