Diploma thesis: Local recognition of reflection graphs on Coxeter groups

This is my diploma thesis, supervised by PD. dr. Ralf Gramlich, as submitted at Technische Universität Darmstadt 31 March 2008 with minor corrections.

The thesis is also available at 0805.2403v1 [math.GR].

Abstract

We provide local recognition results for the reflection graphs on spherical Coxeter groups. In particular, we study the case F₄ which is locally recognizable under additional constraints only. It is then demonstrated in the cases A_n and F₄ how these graph theoretical recognition results can be used to characterize the corresponding Coxeter groups in terms of their reflection centralizers. Finally, we outline the connection to the local recognition of Chevalley groups which is particularly important in the classification of the finite simple groups.

Contents

1. Introduction
   1. Overview
   2. Group theory
   3. Graph theory
   4. Coxeter groups
2. Local recognition of graphs
   1. Some local recognition results
   2. Local recognition of simply laced Weyl graphs
   3. Local recognition of non-simply laced Weyl graphs
   4. Graphs locally like W(F₄)
   5. Recognition results for graphs locally like W(F₄)
3. Group theoretic applications
   1. Recognizing Sym_n
   2. Recognizing Coxeter groups
   3. Outlook: Applications to Chevalley groups
4. Appendix
   1. Computations in SAGE
   2. Computations in GAP

Download:

diploma.pdf