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.

R. Brown, M. Bullejos, T. Porter: Crossed Complexes, Free Crossed Resolutions and Graph Products of Groups, 11--26
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.

C. M. Campbell, P. P. Campbell, B. T. K. Hopson, E. F. Robertson: On the Efficiency of Direct Powers of PGL(2, p), 27--34
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).

D. A. Derevnin, Ann Chi Kim: The Coxeter Prisms in H3, 35--50
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.

D. Hennig, G. Rosenberger: Recent Developments in the Theory of Fuchsian and Kleinian Groups, 51--64
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.

Ann Chi Kim, Yangkok Kim: On Generalized Whitehead Links and 3-Manifolds, 65--72
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.

Jae-Ryong Kim, Moo Ha Woo: Topology Fields and Fixed Points of Flows, 73--82
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.

E. Kudryavtseva, R. Weidmann, H. Zieschang: Quadratic Equations in Free Groups and Topological Applications, 83--122
[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.

A. Mednykh, A. Vesnin: Colourings of Polyhedra and Hyperelliptic 3-Manifolds, 123--132
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.

J. Mennicke: Linear Groups over Rings of Integers, 133--140
[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.

Ch. Menzel, J. R. Parker: Pseudo-Anosov Diffeomorphisms of the Twice Punctured Torus, 141--154
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.

M. Mulazzani: 3-Manifolds with Cyclic Symmetry and (1,1)-Knots, 155--162
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. Szczepanski: Holonomy Groups of Crystallographic Groups with Finite Outer Automorphism Groups, 163--166
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.

K.-I. Tahara: Survey on Dimension Subgroup Problem, 167--182
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.