In the current project, the Principal Investigator will continue his study of the topological structures on which various interesting classes of groups can act. In particular, the PI will show how various types of Artin groups, Coxeter groups, braid groups, Burnside groups, and pure symmetric automorphism groups can be made to act on spaces constructed to have one or more additional properties. The ability to construct such spaces has long been a goal of geometric group theorists. Since many of these groups arise as the fundamental groups of configuration spaces the pattern underlying much of the proposed work can be summarized as follows: a topologically defined space whose fundamental group is of interest is made to act on a piecewise Euclidean cell complex whose structure is understood well enough to have computational comsequences for the group involved. The aim of this project is to increase the number and variety of groups to which thiss type of procedure can be applied.
This type of project continues the recent emphasis within geometric group theory on the practical, computational aspects of mathematics. The results under consideration will not merely show the existence of various objects, but will provide algorithms for their construction and bounds of the complexity of the procedures involved. Thus all of the work described can be implemented as a computer software package which performs these types of computations. This explicit emphasis on the algorithms and their complexity makes geometric group theory a close relative of theoretical computer science and of the real world applications of these techniques.
