Project Summary The study of symmetry in geometrical and abstract contexts is a central issue in such diverse areas as mathematical physics, singularity theory, algebraic geometry, quantum groups and the study of knots and braids. Group theory provides the mathematical framework for the analysis of symmetry. Reflection groups, simple examples of which are the symmetry groups of the five platonic solids, play a key role in all of the areas mentioned above. Thus an improved understanding of reflection groups will significantly enhance the development of several important theories.
	Approach My hope is to use the funding provided by the Australian Research Council to support visits to Sydney by other mathematicians interested in studying reflection groups, to collaborate with me on a variety of related problems (see below). People interested in reflection groups are welcome to contact me: email bobh@maths.usyd.edu.au. I also propose to travel to more international conferences than I have been able to previously.
	Specific objectives Here is a sample of the kinds of problems that may be addressed. Find methods of constructing of W-graphs for irreducible representations of finite Coxeter groups and their associated Iwahori-Hecke algebras. Study automorphism groups of Coxeter groups. If an infinite irreducible Coxeter group has the property that all of its rank 2 parabolic subgroups are finite, does it follow that its full automorphism group is generated by inner automorphisms and graph automorphisms? Rigidity of Coxeter groups: can an infinite irreducible Coxeter group with finite rank 2 parabolics have two non-conjugate Coxeter systems? Discover further extensions of the theory of Coxeter groups from the Euclidean (finite) case to the non-Euclidean (infinite) case. Hyperbolic Coxeter groups are an important special case, of course, but my aim is not to restrict attention to these. In my dreams, I sometimes think that it may be possible to obtain a theory of unitary reflection groups analogous to the theory of Coxeter groups. In all cases, algorithmic aspects of the various problems will be given special attention. In particular, for infinite Coxeter groups one may ask for algorithms that are accessible to finite state automata. Discover, if possible, connections between the formal language of reduced words in W (a Coxeter group) and topological properties of geometrical objects associated with the reflection representation of W.