By Werner Fenchel, Jakob Nielsen, Asmus L. Schmidt
This publication by way of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and complex background. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean aircraft, and this led him - in the course of
World warfare II - to put in writing the 1st chapters of the booklet (in German). whilst Fenchel,
who needed to get away from Denmark to Sweden as a result of German profession,
returned in 1945, Nielsen initiated a collaboration with him on what turned recognized
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) was once accomplished in 1948 and it used to be deliberate to be released within the Princeton
Mathematical sequence. in spite of the fact that, end result of the fast improvement of the topic, they felt
that large alterations needed to be made prior to ebook.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed until eventually
1955), he was once a lot concerned with the foreign association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The files of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity comprise unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and an entire manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The information additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes distinct reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), used to be a great deal concerned with an intensive revision of the curriculum in al-
gebra and geometry, and centred his learn within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly two decades
he additionally positioned a lot attempt into his task as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen venture the way in which he desired to.
After his retirement from the college, Fenchel - assisted through Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came upon time to
finish the booklet ordinary Geometry in Hyperbolic house, which used to be released by way of
Walter de Gruyter in 1989 presently after his dying. at the same time, and with an identical
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal some of the imprecise issues that have been within the unique
manuscript. Fenchel advised me that he pondered elimination components of the introductory
Chapter I within the manuscript, given that this may be coated by means of the e-book pointed out above;
but to make the Fenchel-Nielsen e-book self-contained he eventually selected to not do
so. He did choose to omit
27, entitled Thefundamental crew.
As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed a superb task of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus collage) with the facts examining of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a basic dialogue of the difference to the fashion
of TEX. In so much respects we determined to stick with Fenchel's intentions. although, turning
the typewritten variation of the manuscript into TEX helped us to make sure that the notation,
and the spelling of yes key-words, will be uniform through the ebook. additionally,
we have indicated the start and finish of an evidence within the ordinary variety of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and pride they agreed to submit the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this confident and quickly response. One
particular challenge with the ebook grew to become out to be the copy of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to bring those in ultimate digital shape, yet by way of 1997 it turned transparent that he
would now not be capable to locate the time to take action. notwithstanding, the writer provided an answer
whereby I may still carry detailed drawings of the figures (Fenchel didn't depart such
for Chapters IV and V), after which they might set up the creation of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his advantageous
collaboration about the genuine construction of the figures.
My colleague Bent Fuglede, who has personaHy identified either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally areas the Fenchel-Nielsen manuscript in its right viewpoint. In
this connection i want to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this booklet reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of detailed symbols, an inventory of notation with brief
explanations and connection with the particular definition within the booklet has been incorporated. additionally,
a complete index has been further. In either circumstances, all references are to sections,
We thought of including a whole record of references, yet made up our minds opposed to it as a result of
the overwhelming variety of learn papers during this region. in its place, a miles shorter
list of monographs and different accomplished money owed correct to the topic has been
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this ebook into lifestyles.
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Additional resources for Discontinuous Groups of Isometries in the Hyperbolic Plane
We can consider the set HomC C (ιY, F ) ∈ Ob(Set). 5) HomC C (ι ·, F ) : C 23 Limits is a contravariant functor. 5) is an inverse limit for F . 6) ←− is an isomorphism (a natural equivalence). 7) ←− is an isomorphism for every objects Y of C . 8). 8). Note 7. 4). , ≈ → HomC C (ι lim Fi , F ). HomC (lim Fi , lim Fi ) − ←− ←− ←− For an identity morphism 1lim Fi on the left hand-side, there is ←− α ∈ HomC C (ι lim Fi , F ). 3). Next let −YF : ιY → F be a morphism in C C . 7) there exists a unique element hY ∈ HomC (Y, lim Fi ).
F is an exact functor). Then G takes injective objects of C to injective objects of C . We will prove this assertion as follows. Let I be an injective object of C and let GC 0 φ GC ψ GC G0 be an arbitrary short exact sequence in C . By the assumption, we have the exact sequence G FC 0 Fφ G FC Fψ G FC in C . Since the contravariant functor HomC (·, I ) is an exact functor, we get the exact sequence HomC (F C , I ) (F ψ)∗ ∗ G HomC (F C, I ) (F φ) G HomC (F C , I ) G0 in Ab. 7) in Chapter I). 6) becomes an epimorphism.
IYF = iYF ◦hYY and jFY = jFY ◦hYY for φij : i → j. A terminal object of FnF is said to be an inverse limit (or projective limit or simply limit) of F written as limi∈C F i, or lim Fi . 4) jF commutative. 4) in terms of the notion of a representable functor. First, we will define a functor ι : C C C as follows. f ιf → Y be a morphism of objects Y and Y in C . Then ιY −→ ιY are Let Y − (ιf )i f in C C . , C be a (ιY )(i) = Y and (ιf )(i) = f for every i ∈ Ob(C ). Let F : C functor as before. We can consider the set HomC C (ιY, F ) ∈ Ob(Set).
Discontinuous Groups of Isometries in the Hyperbolic Plane by Werner Fenchel, Jakob Nielsen, Asmus L. Schmidt