206 research outputs found
Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator
A group obtained from a nontrivial group by adding one generator and one
relator which is a proper power of a word in which the exponent-sum of the
additional generator is one contains the free square of the initial group and
almost always (with one obvious exception) contains a non-abelian free
subgroup. If the initial group is involution-free or the relator is at least
third power, then the obtained group is SQ-universal and relatively hyperbolic
with respect to the initial group.Comment: 11 pages. A Russian version of this paper is at
http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V3:
revised following referee's comment
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
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
Infinite presentability of groups and condensation
We describe various classes of infinitely presented groups that are
condensation points in the space of marked groups. A well-known class of such
groups consists of finitely generated groups admitting an infinite minimal
presentation. We introduce here a larger class of condensation groups, called
infinitely independently presentable groups, and establish criteria which allow
one to infer that a group is infinitely independently presentable. In addition,
we construct examples of finitely generated groups with no minimal
presentation, among them infinitely presented groups with Cantor-Bendixson rank
1, and we prove that every infinitely presented metabelian group is a
condensation group.Comment: 32 pages, no figure. 1->2 Major changes (the 13-page first version,
authored by Y.C. and L.G., was entitled "On infinitely presented soluble
groups") 2->3 some changes including cuts in Section
Amenable groups without finitely presented amenable covers
The goal of this article is to study results and examples concerning finitely
presented covers of finitely generated amenable groups. We collect examples of
groups with the following properties: (i) is finitely generated, (ii)
is amenable, e.g. of intermediate growth, (iii) any finitely presented
group with a quotient isomorphic to contains non-abelian free
subgroups, or the stronger (iii') any finitely presented group with a quotient
isomorphic to is large
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
Compact -deformation and spectral triples
We construct discrete versions of -Minkowski space related to a
certain compactness of the time coordinate. We show that these models fit into
the framework of noncommutative geometry in the sense of spectral triples. The
dynamical system of the underlying discrete groups (which include some
Baumslag--Solitar groups) is heavily used in order to construct \emph{finitely
summable} spectral triples. This allows to bypass an obstruction to
finite-summability appearing when using the common regular representation. The
dimension of these spectral triples is unrelated to the number of coordinates
defining the -deformed Minkowski spaces.Comment: 30 page
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