Geometric Group Theory

Just as numbers are the mathematical objects that measure size, groups are the mathematical objects that one needs to describe the symmetries of any mathematical object: no matter what category of objects one is working with, the symmetries (automorphisms) of each object form a group.Having abstracted the notion of a group, it makes sense to study groups as important objects in their own right; thus one has group theory. But in order to elucidate the true nature of an abstract group, one often wants to realise it as the group of automorphisms of specific objects and to use the structure of those objects in order to elucidate the structure of the group. This idea lies at the heart of geometric group theory: typically, given a group, one seeks objects rich in geometric (or other) structure and studies the realisation(action) of the group on this object, using the pwoerful tools of geometry and topology.Conversely, one tries to solve problems in geometry and topology by analysing the groups associated to the spaces at hand, or by encoding well-understood phenomena from group theory into geometric objects.The research to be undertaken here involves both of these approaches:an understanding of the universe of finitely presented groups is sought, largely through an understanding of the geometry and complexity of groups and their actions; on the other hand, attacks on important geometric and topological problems are mounted via group theory.