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Every group is represented in many ways as an epimorphic image of a
free group. It seems therefore futile to search for methods
involving generators and relations which can be used to detect the
structure of a group. Nevertheless, results in the indicated
direction exist. The clue is to ask the right question. Classical
geometry is a typical example in which the factorization of a
motion into reflections or, more generally, of a collineation into
central collineations, supplies valuable information on the
geometric and algebraic structure. This mode of investigation has
gained momentum since the end of last century. The tradition of
geometric-algebraic interplay brought forward two branches of
research which are documented in Parts I and II of these
Proceedings. Part II deals with the theory of reflection geometry
which culminated in Bachmann's work where the geometric information
is encoded in properties of the group of motions expressed by
relations in the generating involutions. This approach is the
backbone of the classification of motion groups for the classical
unitary and orthogonal planes. The axioms in this char acterization
are natural and plausible. They provoke the study of consequences
of subsets of axioms which also yield natural geometries whose
exploration is rewarding. Bachmann's central axiom is the three
reflection theorem, showing that the number of reflections needed
to express a motion is of great importance."
Ever since F. Klein designed his "Erlanger programm", geometries
have been studied in close connection with their groups of
automorphisms. It must be admitted that the presence of a large
automorphismgroup does not always have strong implications for the
incidence-th- retical behaviour of a geometry. For exampl~ O. H.
Kegel and A. Schleiermacher [Geometriae Dedicata 2, 379 - 395
(1974)J constructed a projective plane with a transitive action of
its collineation group on quadrangles, in which, nevertheless every
four points generate a free subplane. However, there are several
important special classes of geometries, in which strong
implications are present. For instance, every finite projective
plane with a doubly transitive collineation group is pappian
(Theorem of Ostrom-Wagner), and every compact connected projective
plane with a flag-transitive group of continuous collineations is a
Moufang plane (H. Salzmann, Pac. J. Math. ~, 217 - 234 (1975)].
Klein's point of view has been very useful for numerous incidence
structures and has established an intimate connection between group
theory and geometry vii P. Plaumann and K. Strambach (eds. ),
Geometry - von Staudt's Point of View, vii-xi. Copyright (c) 1981
by D. Reidel Publishing Company. viii PREFACE 1. 1:1ich is a
guidepost for every modern t:reat:ment of geometry. A few decades
earlier than Klein's proposal, K. G. Ch. von Staudt stated a
theorem which indicates a different point of view and is nowadays
sometimes called the "Fundamental Theorem of Projective Geometry".
When looking for applications of ring theory in geometry, one first
thinks of algebraic geometry, which sometimes may even be
interpreted as the concrete side of commutative algebra. However,
this highly de veloped branch of mathematics has been dealt with in
a variety of mono graphs, so that - in spite of its technical
complexity - it can be regarded as relatively well accessible.
While in the last 120 years algebraic geometry has again and again
attracted concentrated interes- which right now has reached a peak
once more - , the numerous other applications of ring theory in
geometry have not been assembled in a textbook and are scattered in
many papers throughout the literature, which makes it hard for them
to emerge from the shadow of the brilliant theory of algebraic
geometry. It is the aim of these proceedings to give a unifying
presentation of those geometrical applications of ring theo~y
outside of algebraic geometry, and to show that they offer a
considerable wealth of beauti ful ideas, too. Furthermore it
becomes apparent that there are natural connections to many
branches of modern mathematics, e. g. to the theory of (algebraic)
groups and of Jordan algebras, and to combinatorics. To make these
remarks more precise, we will now give a description of the
contents. In the first chapter, an approach towards a theory of
non-commutative algebraic geometry is attempted from two different
points of view.
Every group is represented in many ways as an epimorphic image of a
free group. It seems therefore futile to search for methods
involving generators and relations which can be used to detect the
structure of a group. Nevertheless, results in the indicated
direction exist. The clue is to ask the right question. Classical
geometry is a typical example in which the factorization of a
motion into reflections or, more generally, of a collineation into
central collineations, supplies valuable information on the
geometric and algebraic structure. This mode of investigation has
gained momentum since the end of last century. The tradition of
geometric-algebraic interplay brought forward two branches of
research which are documented in Parts I and II of these
Proceedings. Part II deals with the theory of reflection geometry
which culminated in Bachmann's work where the geometric information
is encoded in properties of the group of motions expressed by
relations in the generating involutions. This approach is the
backbone of the classification of motion groups for the classical
unitary and orthogonal planes. The axioms in this char acterization
are natural and plausible. They provoke the study of consequences
of subsets of axioms which also yield natural geometries whose
exploration is rewarding. Bachmann's central axiom is the three
reflection theorem, showing that the number of reflections needed
to express a motion is of great importance."
When looking for applications of ring theory in geometry, one first
thinks of algebraic geometry, which sometimes may even be
interpreted as the concrete side of commutative algebra. However,
this highly de veloped branch of mathematics has been dealt with in
a variety of mono graphs, so that - in spite of its technical
complexity - it can be regarded as relatively well accessible.
While in the last 120 years algebraic geometry has again and again
attracted concentrated interes- which right now has reached a peak
once more -, the numerous other applications of ring theory in
geometry have not been assembled in a textbook and are scattered in
many papers throughout the literature, which makes it hard for them
to emerge from the shadow of the brilliant theory of algebraic
geometry. It is the aim of these proceedings to give a unifying
presentation of those geometrical applications of ring theo y
outside of algebraic geometry, and to show that they offer a
considerable wealth of beauti ful ideas, too. Furthermore it
becomes apparent that there are natural connections to many
branches of modern mathematics, e. g. to the theory of (algebraic)
groups and of Jordan algebras, and to combinatorics. To make these
remarks more precise, we will now give a description of the
contents. In the first chapter, an approach towards a theory of
non-commutative algebraic geometry is attempted from two different
points of view."
Ever since F. Klein designed his "Erlanger programm", geometries
have been studied in close connection with their groups of
automorphisms. It must be admitted that the presence of a large
automorphismgroup does not always have strong implications for the
incidence-th- retical behaviour of a geometry. For exampl~ O. H.
Kegel and A. Schleiermacher [Geometriae Dedicata 2, 379 - 395
(1974)J constructed a projective plane with a transitive action of
its collineation group on quadrangles, in which, nevertheless every
four points generate a free subplane. However, there are several
important special classes of geometries, in which strong
implications are present. For instance, every finite projective
plane with a doubly transitive collineation group is pappian
(Theorem of Ostrom-Wagner), and every compact connected projective
plane with a flag-transitive group of continuous collineations is a
Moufang plane (H. Salzmann, Pac. J. Math. ~, 217 - 234 (1975)].
Klein's point of view has been very useful for numerous incidence
structures and has established an intimate connection between group
theory and geometry vii P. Plaumann and K. Strambach (eds. ),
Geometry - von Staudt's Point of View, vii-xi. Copyright (c) 1981
by D. Reidel Publishing Company. viii PREFACE 1. 1:1ich is a
guidepost for every modern t:reat:ment of geometry. A few decades
earlier than Klein's proposal, K. G. Ch. von Staudt stated a
theorem which indicates a different point of view and is nowadays
sometimes called the "Fundamental Theorem of Projective Geometry".
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