Ebook: Generators and Relations in Groups and Geometries
- Tags: Group Theory and Generalizations, Field Theory and Polynomials, Linear and Multilinear Algebras Matrix Theory, Differential Geometry
- Series: NATO ASI Series 333
- Year: 1991
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xv
Classical Groups....Pages 1-45
Generators of Automorphism Groups of Modules....Pages 47-67
Generators of Automorphism Groups of Cayley Algebras....Pages 69-94
Products of Matrices....Pages 95-123
Reflection groups....Pages 125-164
Unitary geometry....Pages 165-177
Lie and Algebraic Johnsen Groups....Pages 179-193
2-Generation of finite simple groups and some related topics....Pages 195-233
Coxeter Groups and three Related Topics....Pages 235-278
Geometric structure of conjugacy classes in algebraic groups....Pages 279-290
Groups with Polynomial Growth and Differential Geometry....Pages 291-320
Analyticity and Growth of Pro-p-Groups....Pages 321-341
Intersections of Local Algebraic Extensions of a Hilbertian Field....Pages 343-405
Generators and Relations for Discontinuous Groups....Pages 407-436
Back Matter....Pages 437-447
Content:
Front Matter....Pages i-xv
Classical Groups....Pages 1-45
Generators of Automorphism Groups of Modules....Pages 47-67
Generators of Automorphism Groups of Cayley Algebras....Pages 69-94
Products of Matrices....Pages 95-123
Reflection groups....Pages 125-164
Unitary geometry....Pages 165-177
Lie and Algebraic Johnsen Groups....Pages 179-193
2-Generation of finite simple groups and some related topics....Pages 195-233
Coxeter Groups and three Related Topics....Pages 235-278
Geometric structure of conjugacy classes in algebraic groups....Pages 279-290
Groups with Polynomial Growth and Differential Geometry....Pages 291-320
Analyticity and Growth of Pro-p-Groups....Pages 321-341
Intersections of Local Algebraic Extensions of a Hilbertian Field....Pages 343-405
Generators and Relations for Discontinuous Groups....Pages 407-436
Back Matter....Pages 437-447
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