Sketched stable planes [Elektronische Ressource] / vorgelegt von Anke Wich
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Sketched stable planes [Elektronische Ressource] / vorgelegt von Anke Wich

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Sketched Stable PlanesVon der Fakultat Mathematik und Physik der Universitat Stuttgart zur Erlangung der Wurde einer Doktorin derNaturwissenschaften (Dr. rer. nat.) genehmigte Abhandlungvorgelegt vonAnke Wichaus Kelkheim im TaunusHauptberichter Prof. Dr. Markus StroppelMitberichter Prof. Dr. Hermann HahlTag der mundlichen Prufung 13. Februar 2003 Institut fur Geometrie und Topologie der Universitat Stuttgart2003Anke WichInstitut fur Geometrie und TopologieLehrstuhl fur GeometrieUniversitat StuttgartPfa enwaldring 57D-70569 Stuttgartwich@mathematik.uni-stuttgart.deMathematics Subject Classi cation (MSC 1991) :51H10 Topological linear incidence geometries51H20 Topological geometries on manifolds51A40 Translation planes and spreads51A45 Incidence structures imbeddable into projective geometries51A10 Homomorphism, automorphism and dualities22F50 Groups as automorphisms of other structures57S20 Noncompact Lie groups of transformations51J99 Incidence groupsKeywords : topological geometry, stable plane, complex projective plane, transformationgroup, group partition, embeddingThis thesis is also available as an online publication athttp://elib.uni-stuttgart.de/opusAbstractStandard objects in classical (topological) geometry are the real a ne and hyperbolicplanes. Both of them can be seen as (open) subplanes of the real projective plane(endowed with the standard topology) and thus share a common theory.

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Sketched Stable Planes
Von der Fakultat Mathematik und Physik der Universitat Stuttgart
zur Erlangung der Wurde einer Doktorin der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von
Anke Wich
aus Kelkheim im Taunus
Hauptberichter Prof. Dr. Markus Stroppel
Mitberichter Prof. Dr. Hermann Hahl
Tag der mundlichen Prufung 13. Februar 2003
Institut fur Geometrie und Topologie der Universitat Stuttgart
2003Anke Wich
Institut fur Geometrie und Topologie
Lehrstuhl fur Geometrie
Universitat Stuttgart
Pfa enwaldring 57
D-70569 Stuttgart
wich@mathematik.uni-stuttgart.de
Mathematics Subject Classi cation (MSC 1991) :
51H10 Topological linear incidence geometries
51H20 Topological geometries on manifolds
51A40 Translation planes and spreads
51A45 Incidence structures imbeddable into projective geometries
51A10 Homomorphism, automorphism and dualities
22F50 Groups as automorphisms of other structures
57S20 Noncompact Lie groups of transformations
51J99 Incidence groups
Keywords : topological geometry, stable plane, complex projective plane, transformation
group, group partition, embedding
This thesis is also available as an online publication at
http://elib.uni-stuttgart.de/opusAbstract
Standard objects in classical (topological) geometry are the real a ne and hyperbolic
planes. Both of them can be seen as (open) subplanes of the real projective plane
(endowed with the standard topology) and thus share a common theory. This may serve
as a brief illustration of the importance of the notion of embeddability.
One particularly nice class of topological planes are the so called stable planes –in
fact, the above examples are stable planes; as well as the projective planes over the
real and complex numbers, Hamilton quaternions and Cayley octaves, the so called
classical planes. Moreover, every open subplane of a stable plane again is a stable plane.
Consequently, one way of understanding a given stable plane is trying to embed it into
one of more profound acquaintanceship, preferredly one of the classical planes.
An elegant way of constructing stable planes uses stable partitions of Lie groups.
Planes of that type can be treated more eciently studying these groups along with
certain stabilisers, the so called sketches, rather than the original geometries. This
method has so far yielded results in several cases where intrinsic methods had not been
gratifying.
Maier in his dissertation gives a classi cation of all 4-dimensional connected Lie groups
which allow for a stable partition. Only one of them, the Frobenius group = RnHei R,3
had not been expected, and it hosts an in nite number of stable partitions. Our objective
is whether or not the resulting stable planesP are embeddable into an already well known
plane. Using sketches, it can be proved that none of these planesP is embeddable into
the classical projective planeP C. As an interesting counterpoint, those planes — hostile2
as they are towards being embedded into classical planes — do contain an abundance
of both, a ne and non-a ne 2-dimensional classical subplanes.
The full automorphism group of such a planeP does not contain a certain selection
of classical groups. Some conclusions can be drawn as to how soluble is : either it
is soluble or it contains one copy of a subgroup with Lie algebra sl R. The normaliser2
N ( )of in turns out to be soluble, after all.

On a more general basis, the interplay of being a sketched geometry and a stable plane
is studied : Is there any particular reason why all the examples of sketched stable planes
so far have been point homogeneous geometries ? And indeed, any line homogeneous
sketched stable plane is necessarily ag homogeneous.
iZusammenfassung
DerBegri der stabilen Ebenen verallgemeinert alltagliche klassische Ebenen wie die re-
elle a ne Ebene oder die reelle hyperbolische Ebene. Besonders sch one Exemplare lassen
sich aus Gruppen mit gewissen Partitionen konstruieren, die sogenannten skizzierten sta-
bilen Ebenen. Die Gruppen, die solche stabilen Partitionen zulassen, sind sehr hau g
Liegruppen und haben nach einem Satz von Lowen die Dimension 2, 4, 8 oder 16.
Maier klassi zierte alle vierdimensionalen Liegruppen mit stabilen Partitionen. Die
stabilen Ebenen, die sich aus den vier Kandidaten ergeben, sind wohlbekannt — mit
Ausnahme derer, die aus der Frobeniusgruppe =RnHei R entstehen. Diese Familie3
vonEbenenwirdhiern aher beleuchtet.
Neben dem erwahn ten Konstruktionsmechanismus spielt der Begri der Einbettung
eine tragende Rolle. Beispielsweise lassen sich die a ne und auch die hyperbolische reelle
Ebene als oene Unterebenen einbetten in die reelle projektive Ebene, erschlie en sich
mithin dem gemeinsamen Zugri mit Hilfe nur einer Theorie. Umgekehrt ist jede o ene
Unterebene einer stabilen Ebene wiederum eine stabile Ebene. Auf diesem Wege kann
man sich also mit einer fremden stabilen Ebene vertraut machen — indem man namlich
eine bekannte Ebene ndet, in die sie einbettbar w are. Die begehrtesten “Betten” sind
naturlich die klassischen stabilen Ebenen, also die projektiven Ebenen uber den reellen
Zahlen, den komplexen Zahlen, den Hamiltonschen Quaternionen und den Cayleyschen
Oktaven.
Es wird nachgewiesen, da keine der aus konstruierten stabilen Ebenen auf irgendei-
nem Wege in die vierdimensionale komplexe projektive Ebene eingebettet werden kann.
Dieses Ergebnis schrankt die Suche nach der vollen Automorphismengruppe einer
solchen Ebene deutlich ein : gewisse klassische Gruppen konnen nicht als Automor-
phismengruppen auftauchen, und mithin ist entweder selbst au osbar oder enthalt
genau ein Exemplar einer Untergruppe mit Liealgebra sl R. Ihr Normalisator N ( )ist2
au osbar.
Umgekehrt ergibt sich, da diese Ebenen selber eine Vielzahl von zweidimensionalen
Unterebenen enthalten, die ane oder nichta ne Unterebenen der reellen a nen Ebene
sind.
In allgemeinerem Kontext wird ausgeleuchtet, weshalb bislang keine anderen als punkt-
homogene skizzierte stabile Ebenen bekannt sind: jede geradenhomogene skizzierte sta-
bile Ebene ist notwendigerweise bereits fahnenhomogen.
ii



Contents
Abstract i
Preface v
Kurzfassung in deutscher Sprache ix
1. Foundations 1
1.1. Sketched geometries.............................. 1
1.1.1. Categories and sketched geometries .. 1
1.1.2. Homogeneity and sketched geometries..... 5
1.2. Stable planes ....... 7
1.3. Morphisms and embeddings of stable planes ................ 8
1.4. Construction of stable planes from stable partitions ........ 10
1.5. Stable partitions of 4-dimensional Lie groups ..... 16
2. Line homogeneous sketched stable planes 19
2.1. Euclidean, hyperbolic and skew hyperbolic geometries ........... 19
2.2. Non-isotropic points therein ..................... 34
2.3. Classi cation of line homogeneous sketched stable planes ......... 39
3. A non-embeddability theorem for Peter planes 49
3.1. The planes ... ................................. 49
3.2. ... and their bed...... 52
3.3. A categorical user’s manual for the embedding of planes.......... 54
3.3.1. Transition from incidence structures to geometries ..... 56
3.3.2. Excursus on the topologies involved ................. 60
3.3.3. Transition from geometries to sketched geometries ..... 65
3.3.4. Transition from sketched geometries to sketches .......... 67
3.4. Hunting down group monics .......... 68
3.5. The point orbits................................ 7
3.6. The point stabilisers ... 82
3.7. The line stabilisers ........ 84
3.8. One more way of not embedding Peter planes 8
iiiContents
4. Classical subplanes in Peter planes 91
4.1. Two prototypes ................................ 91
4.2. Sketched Baer subplanes from 2-dimensional Lie subalgebras ....... 96
4.3. The prototypes as sketched Baer subplanes of the original Peter planes . 100
4.4. Classi cation of 2-dimensional Lie subalgebras of g ..103
4.5. Abelian bres in stable partitions of g....................109
4.6. A ne lines in Peter planes ................111
5. On the automorphism group of Peter planes 115
5.1. is compact-free ....................15
5.1.1. The commutator subgroup of a compact connected Lie group...17
5.1.2. The centre of a compact connected Lie group .120
5.1.3. Simply connected compact Lie groups......122
5.2. Some groups the automorphism group does not contain ..........125
5.2.1. SO R is not an automorphism group ofP ...1263
5.2.2. SL C is not an am ofP....1272
5.2.3. SU C is not an am group ofP ...1272
5.2.4. Application of a result by Bickel..................128
5.3. How soluble is the automorphism group ?129
5.3.1. Zoological considerations concerning so R ...1293
5.3.2. Consequences for the Levi decomposition ...13
5.3.3. Semisimple complex Lie algebras, real and compact forms.....134
5.3.4. What the classi cation of simple Lie algebras can do for us ....137
5.3.5. Solubility revisited ..........................140
5.3.6.y of a normaliser..140
A. Appendix 147
A.1. Topology....................................147
A.2. Groups and topological groups...149
A.3. Lie algebras and Lie groups ....149
Bibliography 153
Index of Symbols 158
Index of Subjects 159
ivPreface
Mathematics has a long and fruitful tradition of translating problems from one of its
areas into another one which has already been understood more deeply and which might
shed a new light on the subject at hand. The translation mechanism which will be the
central thread running through the present thesis is called sketching, and it helps to
understand problems dealing with geometries by studying their transformation groups
along with suitable stabilisers.
It is as early as 1927 that Young in [77] introduces group partitions of possibly non-
abelian groups. In 1954, this notion is taken up by Andre [2] who uses them for the
construction of certain point homogeneous geometries, his so-called “translation struc-
tures”. (In our notation, thus, he treats incidence structures of the form P ( ;{1},F),
whereF is a group partition of the abstract group .) Andre characterises translation
planes as being precisely those translation structures which arise from planar partitions
(“congruences”) of a necessarily abelian group. In 1951, Freudenthal in a little aside
in [13] hints at the possibility of constructing ag homogeneous geometries on planes
from a group along with two of its subgroups. The same train of thought is developed
in 1961 by Higman and McLaughlin in [22].
Stroppel [59] in 1992 gives a useful generalisation of the method to geometries on
planes which are not ag homogeneous. Finally in 1993, Stroppel [60] takes a cat-
egorical point of view and introduces a reconstruction method for geometries with an
arbitrarynumberoftypes(point,lines,...). Moreover,heestablishesthatthemethod
is a reconstruction method, indeed : any geometry satisfying suitable homogeneity con-
ditions, a so-called sketched geometry, is fully determined by its transformation group
along with certain stabilisers, its so-called sketch.
Applications of this translation technique have been highly rewarded in quite a number
of cases where mere study of the geometrical problems had not been successful. A beau-
tiful recent example is Grundhofer, Kramer and Knarr’s classi cation [ 16]+[17]
of ag homogeneous compact connected polygons. Further success could be noted in
[57], [61], [63], [65]and[72].
The present thesis wishes to add to the applications of Stroppel’s reconstruction
method to certain “topological planes”. Topological geometry studies incidence ge-
ometries which are endowed with a topology that in a certain sense is compatible with
the geometric operations. Very little can be said about topological planes in general,
though, and they may escape far from the scope of classical planes. It is thus desirable
to impose further topological or homogeneity conditions.
v
Preface
Salzmann and his school have been primarily concerned with the classi cation of
compact connected projective planes, where the key is given by the dimension of their
automorphism group. Yet, other types of topological planes arose to the left and right
of their way, one of which will be studied here : In 1976, Lowen [31] coined the no-
tion of stable planes, i.e., topological linear spaces where planarity was modelled by
an additional “stability axiom”. It in particular covers the classical projective, a ne
and hyperbolic planes. Additional topological hypotheses – local compactness and -
nite positive topological dimension – stimulate surprisingly strong results on their actual
dimension and also on their automorphism groups; see [39], [32].
Every open subplane of a stable plane is a stable plane on its own. For instance, the
real a ne plane A R and the real hyperbolic plane IHR are both open subplanes of2
the real projective planeP R and thus share a common theory. This is why, given an2
unknown example of the species, it is an obvious question if it could be established as
an open subplane of some well-known, preferredly classical stable plane. In that way,
embeddability problems mark their appearance on stage. And it is here that things come
to full circle : Stroppel’s mechanism of translating di cult problems on geometries
into a corresponding question on their sketches has also lead the way towards quite
rewarding results on embeddability. Stroppel [61], for instance, achieves an embedding
of Strambach’s 2-dimensional SL R-plane into Lowen’s 4-dimensional SL C-plane.2 2
We will join a brief guide to the actual parts of the present thesis. The general structure
is thus that Chapter 1 provides for the fundamental notions and results on sketched
geometries and stable planes. Chapter 2 is independent of all the other parts and deals
with the general theory of sketched stable planes, whereas the remaining chapters treat
one particular family of sketched stable planes. Chapters 3 and 4 are parallel studies of
embeddability problems. Chapter 5 relies upon the results from chapter 3 and interprets
their relevance for the study of the automorphism group of the planes under review.
Chapter 2 asks for the interplay of being a stable plane and being a sketched geometry.
A sketched linear space is necessarily point or line homogeneous. Yet a huge dominance
of the point homogeneous species has been observed, and for a good reason : it can be
proved that a line homogeneous sketched stable plane must be ag homogeneous. The
proof is based on Lowen’s classi cation [ 38] of stable planes with at least two isotropic
points, which yields a list of candidates. In fact, a line homogeneous sketched stable
plane ( ,P) entirely consists of -isotropic points. Some of Lowen’s candidates will
be disquali ed due to possession of non-isotropic points. For the remaining ones ag
homogeneity can be established.
Chapters 3 through 5 take up the main issue of the present thesis : Peter planes. Such
were baptised those stable planes which arise from stable partitions of the Lie group
= RnHei R, stemming from Maier’s classi cation [ 44] of 4-dimensional connected3
Lie groups allowing for stable partitions. Maier proves that there are four such groups,
and the stable planes three of them give rise to are well-known. It is the fourth group
and the corresponding stable planes which have so far evaded any rm grip.
After a brief introduction to the subject, Chapter 3 deals with the question whether
vi