Generators and Relations in Groups and Geometries

1. Front Matter

   Pages i-xv

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2. Optimal factorization of matrices, length problems

   1. Classical Groups

      • Erich W. Ellers

      Pages 1-45

   2. Generators of Automorphism Groups of Modules

      • H. Ishibashi

      Pages 47-67

   3. Generators of Automorphism Groups of Cayley Algebras

      • Huberta Lausch

      Pages 69-94

   4. Products of Matrices

      • Thomas J. Laffey

      Pages 95-123

3. Reflection geometry

   1. Reflection groups

      • Frieder Knüppel

      Pages 125-164

   2. Unitary geometry

      • Martin Götzky

      Pages 165-177

   3. Lie and Algebraic Johnsen Groups

      • Peter Plaumann

      Pages 179-193

4. Nice generators and relations, applications

   1. 2-Generation of finite simple groups and some related topics

      • L. Di Martino, M. C. Tamburini

      Pages 195-233

   2. Coxeter Groups and three Related Topics

      • Arjeh M. Cohen

      Pages 235-278

   3. Geometric structure of conjugacy classes in algebraic groups

      • T. A. Springer

      Pages 279-290

   4. Groups with Polynomial Growth and Differential Geometry

      • E. Musso, F. Tricerri

      Pages 291-320

   5. Analyticity and Growth of Pro-p-Groups

      • A. Caranti

      Pages 321-341

   6. Intersections of Local Algebraic Extensions of a Hilbertian Field

      • Moshe Jarden

      Pages 343-405

   7. Generators and Relations for Discontinuous Groups

      • Bruno Zimmermann

      Pages 407-436

5. Back Matter

   Pages 437-447

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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.

• Algebraic structure
• algebra
• algebraic group
• automorphism
• differential geometry
• matrix theory

• Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

  A. Barlotti

• Department of Mathematics, University of Toronto, Toronto, Canada

  E. W. Ellers

• Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany

  P. Plaumann, K. Strambach

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