Introduction: groups and geometries

Every group is supposed to act upon a certain set preserving some additional structure. This set, which is almost always a space, should be associated to the group in a natural way – in many cases, the set comes first and the group is associated to the set. Linear groups act on vector spaces. Fuchsian groups act on the hyperbolic plane. Symmetry groups of geometric configurations act upon these objects.

In some cases the right space to act on is not that easily found. The mapping class group of a surface, i.e., the group of homotopy classes of homeomorphisms of that surface does not act upon the surface since it is a proper quotient of its automorphism group. Nevertheless, it acts upon the Teichmüller space of the surface. The group of outer automorphisms of a free group of finite rank acts on the Culler-Vogtmann space (outer space). These spaces have been christened in honor of those who found them. This indicates that it is not at all easy to find the right space.

How can we distinguish the right action from other actions? Most groups admit actions on many spaces. What distinguishes a good choice? As a rule of thumb, we will aim at small stabilizers as well as a small quotient space. In the case of a free action, every orbit is isomorphic to the group, and if the quotient space is a point, then there is only one orbit to deal with.