Ebook: Discontinuous Groups of Isometries in the Hyperbolic Plane

This book by Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a long and complicated history. In 1938-39, Nielsen gave a series of lectures on
discontinuous groups of motions in the non-euclidean plane, and this led him - during
World War II - to write the first two chapters of the book (in German). When Fenchel,
who had to escape from Denmark to Sweden because of the German occupation,
returned in 1945, Nielsen initiated a collaboration with him on what became known
as the Fenchel-Nielsen manuscript. At that time they were both at the Technical
University in Copenhagen. The first draft of the Fenchel-Nielsen manuscript (now
in English) was finished in 1948 and it was planned to be published in the Princeton
Mathematical Series. However, due to the rapid development of the subject, they felt
that substantial changes had to be made before publication.
When Nielsen moved to Copenhagen University in 1951 (where he stayed until
1955), he was much involved with the international organization UNESCO, and the
further writing of the manuscript was left to Fenchel. The archives of Fenchel now
deposited and catalogued at the Department of Mathematics at Copenhagen Univer-
sity contain two original manuscripts: a partial manuscript (manuscript 0) in Ger-
man containing Chapters I-II (

I -15), and a complete manuscript (manuscript I) in
English containing Chapters I-V (

1-27). The archives also contain part of a corre-
spondence (first in German but later in Danish) between Nielsen and Fenchel, where
Nielsen makes detailed comments to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen University in 1956 (and stayed there
until 1974), was very much involved with a thorough revision of the curriculum in al-
gebra and geometry, and concentrated his research in the theory of convexity, heading
the International Colloquium on Convexity in Copenhagen 1965. For almost 20 years
he also put much effort into his job as editor of the newly started journal Mathematica
Scandinavica. Much to his dissatisfaction, this activity left him little time to finish the
Fenchel-Nielsen project the way he wanted to.
After his retirement from the university, Fenchel - assisted by Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - found time to
finish the book Elementary Geometry in Hyperbolic Space, which was published by
Walter de Gruyter in 1989 shortly after his death. Simultaneously, and with the same
collaborators, he supervised a typewritten version of the manuscript (manuscript 2) on
discontinuous groups, removing many of the obscure points that were in the original
manuscript. Fenchel told me that he contemplated removing parts of the introductory
Chapter I in the manuscript, since this would be covered by the book mentioned above;
but to make the Fenchel-Nielsen book self-contained he ultimately chose not to do
so. He did decide to leave out
27, entitled Thefundamental group.



As editor, I started in 1990, with the consent of the legal heirs of Fenchel and
Nielsen, to produce a TEX-version from the newly typewritten version (manuscript 2).
I am grateful to Dita Andersen and Lise Fuldby-Olsen in my department for hav-
ing done a wonderful job of typing this manuscript in AMS- TEX. I have also had
much help from my colleague J0rn B0rling Olsson (himself a student of Kate Fenchel
at Aarhus University) with the proof reading of the TEX-manuscript (manuscript 3)
against manuscript 2 as well as with a general discussion of the adaptation to the style
of TEX. In most respects we decided to follow Fenchel's intentions. However, turning
the typewritten edition of the manuscript into TEX helped us to ensure that the notation,
and the spelling of certain key-words, would be uniform throughout the book. Also,
we have indicated the beginning and end of a proof in the usual style of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my great relief and satisfaction they agreed to publish the manuscript in their series
Studies in Mathematics. I am most grateful for this positive and quick reaction. One
particular problem with the publication turned out to be the reproduction of the many
figures which are an integral part of the presentation. Christian Siebeneicher had at
first agreed to deliver these in final electronic form, but by 1997 it became clear that he
would not be able to find the time to do so. However, the publisher offered a solution
whereby I should deliver precise drawings of the figures (Fenchel did not leave such
for Chapters IV and V), and then they would organize the production of the figures in
electronic form. I am very grateful to Marcin Adamski, Warsaw, Poland, for his fine
collaboration concerning the actual production of the figures.
My colleague Bent Fuglede, who has personaHy known both authors, has kindly
written a short biography of the two of them and their mathematical achievements,
and which also places the Fenchel-Nielsen manuscript in its proper perspective. In
this connection I would like to thank The Royal Danish Academy of Sciences and
Letters for allowing us to include in this book reproductions of photographs of the two
authors which are in the possession of the Academy.
Since the manuscript uses a number of special symbols, a list of notation with short
explanations and reference to the actual definition in the book has been included. Also,
a comprehensive index has been added. In both cases, all references are to sections,
not pages.
We considered adding a complete list of references, but decided against it due to
the overwhelming number of research papers in this area. Instead, a much shorter
list of monographs and other comprehensive accounts relevant to the subject has been
collected.
My final and most sincere thanks go to Dr. Manfred Karbe from Walter de Gruyter
for his dedication and perseverance in bringing this publication into existence.