|Geometric group theory is relatively young field but with older and deeper roots in which groups were studied from combinatorial and topological perspectives. Combinatorial group theory traces back to the work of Dehn, and focuses on the combinatorial nature of cell complexes associated to groups. Topological methods in group theory focused on the cohomology of groups and their finiteness properties, and hence viewed groups as essentially topological ob jects. In the mid 1980's, spurred by ideas of Jim Cannon and Misha Gromov, group theorists began to pay attention to the geometric structures which cell complexes can carry. This attention shed a great deal of light on the earlier combinatorial and topological investigations into group theory, and stimulated other innovative ideas which have been developing at a rapid pace. As it has grown over the past 20 years, geometric group theory has developed many different facets, including geometry, topology, analysis, logic.
These facets are often studied in the context of specific groups or classes of groups: Artin groups, Coxeter groups, braid groups, mapping class groups, the Torrelli group, Out(Fn ), Aut(Fn ), lattices in Lie groups, square-complex groups, Thompson's group, automata groups etc.
The new, more geometric, perspectives have enabled rapid progress on many of these fronts. A tremendous solidification of previously disparate results has also occurred. This semester program at MSRI will capitalize on this recent surge of activity. We will bring people from the various branches of geometric group theory together to work on some of the many longer-standing open questions in the field that are now being studied from fresh and promising perspectives, and to further strengthen the connections the field has to the other branches of mathematics.|