Research and Exposition in Mathematics
Volume 27
J. Mennicke, Jung Rae Cho (eds.)
Group Theory and Low-Dimensional Topology.
German-Korean Workshop, Pusan 2000
200 p., soft cover, ISBN 3-88538-227-X, EUR 30.00, 2003
Contents with Abstracts
P. Ackermann, M. Näätänen, G. Rosenberger:
The Arithmetic Fuchsian Groups with Signature (0; 2, 2, 2, q), 1--10
- We determine the arithmetic Fuchsian groups with signature (0; 2, 2, 2, q)
in terms of corresponding trace triples. In the case when q is even we use
the fact that there is a one-to-one correspondence between these groups and
Fuchsian groups with signature (1; q/2) containing these. In the case when q
is odd the corresponding groups are generated by two elements and we use
Takeuchi's characterization of Fuchsian groups in terms of the traces of the
elements.
- The category of crossed complexes gives an algebraic model of
CW-complexes and cellular maps. Free crossed resolutions of groups
contain information on a presentation of the group as well as higher
homological information. We relate this to the problem of calculating
non-abelian extensions. We show how the strong properties of this
category allow for the computation of free crossed resolutions of
graph products of groups, and so obtain computations of higher
homotopical syzygies in this case.
- The paper proves the efficiency of direct powers of the group
G(p) given by the presentation < a , b|a2 , bp ,
(ab2)4 , (abab2)3 > . Depending
on properties of the prime p, the group G(p) is either PGL(2, p) or
C2 x PSL(2, p).
- We give combinatorial and metrical descriptions of the Coxeter n-gon
prisms in H3, starting out from the work of E. M. Andreev. As
applications, we obtain the volumes of the prisms.
- The theory of Fuchsian and Kleinian groups and their orbit spaces, Riemann
surfaces and hyperbolic 3-folds, is a deep and beautiful subject with many
important applications in analysis, geometry, low dimensional topology,
combinatorial group theory and number theory. We have made no attempt here
to present a general survey of the theory of Fuchsian and Kleinian groups,
preferring instead to concentrate on our own and related work on these groups.
We recommend the books "The geometry of discrete groups" by A. F. Beardon
[Graduate Texts in Mathematics 91, Springer-Verlag 1983] and "Kleinian groups"
by B. Maskit [Grundlehren Math. Wissenschaften, Springer-Verlag 1988] as good
places to begin for readers who want to learn more about Fuchsian and Kleinian
groups.
This paper is intended as a survey of some recent work on Fuchsian and
Kleinian groups. Here is an outline of the topics we intend to cover:
(1) Fuchsian and Kleinian groups.
(2) Universal constraints for Fuchsian and Kleinian groups. Two-generator
discrete groups.
(3) Arithmetic Fuchsian and Kleinian groups.
(4) Generalized triangle and tetrahedron groups.
- We construct an infinite family of closed (hyperbolic) orientable
3-manifolds M(2m+1, n, k) by pairwise identifications of faces in the
boundary of certain polyhedral 3-balls. We prove that these manifolds
are n-fold cyclic branched coverings of the 3-sphere over a link
Wm-1. Our works generalize the Whitehead link, due to Helling,
Kim and Mennicke as a special case m = 2.
- Topology describes structure patterns on a given set. The family of
all topologies given on a set form a complete lattice. The purpose of
this study is to introduce a topology field which is a map from a given
phase space to the lattice of all topologies on the given space. Next
we describe some properties concerning a pattern of relationship between
topological transformation groups (flows) and topology fields.
- [Abstract-pdf]
The fundamental groups of closed surfaces admit canonical
one-relator presentations where the defining relation $\Pi_*$ is either a
product of commutators or a product of squares of the generators; hence
it is a word in which each generator appears exactly twice, namely twice with
exponent $\pm 1$ or once with exponent $\pm 2$. Words of this type
are called quadratic words.
One way to investigate these groups is to study quadratic words
$W(z_1,\dots, z_n)$
in a free group $F = \langle a_1,b_1, \dots, a_g, b_g \,|\,-\rangle$ or
$F = \langle a_1, \dots, a_h \,|\,-\rangle$ and to look for
solutions in $F$ of equations like
$$
W(z_1, \dots, z_n) = \cases{
1 &\cr
\prod_{j=1}^g [a_j,b_j]&\cr
\prod_{j=1}^h a_j^2 .&}
$$
The solutions of these equations are described in the following sections,
where they are also related to results in $2$- and $3$-dimensional
topology. A final section contains an introduction to coincidence theory
and applications of it to quadratic equations in free groups where
the right side is more complicated than above. We provide generalizations
or new proofs of known results.
- We survey recent results on hyperelliptic 3-manifolds whose fundamental
groups are finite index subgroups in Coxeter groups. These fundamental groups
correspond to colourings (in particular, to the classical regular four-colouring)
as well as hamiltonian cycles (and their generalizations) of Coxeter polyhedra.
- [Abstract-pdf]
This paper is concerned with an extention of the classical congruence
subgroup theorem for the subgroup $SL_n (\mathbb{Z}), n \geq 3$.
{\bf Theorem:} Fix $m \in \mathbb{N}$, and consider the subgroup
generated by the matrices
$Q_m = < I + ml_{ij}, \forall i,j = 1, \dots n, i + j >.$
$Q_m$ is of finite index in $SL_n \mathbb{Z}$, and it can be characterised
by congruences.
\bigskip
In a second part, we report on some work on the congruence subgroup
problem for cocompact groups, in particular for groups in quarternion
algebras. This is work in progress.
- The group of isotopy classes of diffeomorphisms from a surface of finite
type to itself (otherwise known as the mapping class group) is a familiar
object. There are two fundamental theorems which enable one to describe
elements of this group. The first theorem is due to Nielsen and says (in
the later reformulation of Thurston) that surface diffeomorphisms may be
classified as (i) periodic, (ii) reducible or (iii) pseudo-Anosov. The
second theorem is due to Dehn and says that the mapping class group is
finitely generated by elementary diffeomorphisms called Dehn twists.
There are various algorithms for deciding whether a given diffeomorphism is
periodic, reducible or pseudo-Anosov. In particular, there are algorithms
due to Bestvina and Handel and Hamidi-Tehrani and Chen. The latter algorithm
uses the piecewise linear action of the mapping class group on the piecewise
linear structure of projective measured lamination space given by the p 1-train tracks of Birman and Series. This
piecewise linear structure and piecewise linear action was worked out in
detail for the twice punctured torus by Parker and Series. The purpose of
this note is to use this description to give an algorithm which takes a
particular diffeomorphism specified as a word in a given set of Dehn twist
generators and decides whether or not it is pseudo-Anosov. Thus it can be
thought of as a realisation of part of the Hamidi-Tehrani and Chen algorithm
in this case.
It is known that the mapping torus of a pseudo-Anosov diffeomorphism is a
hyperbolic 3-manifold. However there are very few descriptions in the
literature of concrete examples of such manifolds. An application of our
method is that we can construct many examples of hyperbolic 3-manifolds
which fibre over the circle with fibre the twice punctured torus.
In particular, we can construct the Whitehead link complement in this
way using one of the simplest pseudo-Anosov diffeomorphisms. We work
this example out in detail in the last section.
- The connections between the 3-manifolds with cyclically presented
fundamental groups and the strongly-cyclic branched coverings of
(1, 1)-knots are studied. A list of recent results on the topics
is shown, including some important examples.
- A crystallographic group G of dimension
n is a discrete, cocompact subgroup of isometries of euclidean space
Rn. From Bieberbach's theorems any such group contains a
free abelian subgroup Zn of finite index. Moreover the
finite group G = G / Zn acts
faithfully by conjugation on Zn. We shall call the
correspond representation G --> GL(n, Z) the holonomy
representation of G and G a holonomy
group of G . A Bieberbach group is a
torsion free crystallographic group.
In a previous paper [Bull. Belg. Math. Soc. 3 (1996) 585--593] the
author proved that the outer automorphism group of a crystallographic
group G is finite if and only if in the
holonomy representation of G all Q-irreducible
components are multiplicity free and R-irreducible.
In this paper he proves that the permutation groups Sn,
n <= 6 are holonomy groups of a Bieberbach groups with finite
outer automorphism group. Moreover the relation between, the class
of finite groups with finite Whitehead group and the class of finite
groups which are holonomy groups of Bieberbach groups with finite
outer automorphism groups are given.
- Let G be a group, and g n(G)
(n >= 1) denote the n-th term of the lower central series of G.
Let ZG be the integral group ring of G, and Dn(G) the n-th dimension
subgroup of G, namely Dn(G) the intersection of G and (1 +
D (G)n with the augmentation ideal
D (G).
The dimension subgroup problem is solved in an almost complete form, namely
exp(Dn(G) / g n(G) at most
2 for any group G and n >= 1, and there exists some group G(n) such that
exp(Dn(G(n))) = 2 for any n >= 4. And so we offer some
generalizations of the dimension subgroup problem. One of them is the
generalized dimension subgroup problem, and the other is the Fox subgroup
problem.