Algebra Seminar talk
Some Notes on Wild Geometric Group Theory
Classical geometric group theory can be described as the study of actions of groups on spaces which preserve certain structures. In particular the notion of a cocompact proper action is key in relating information between finitely presented groups and CW-complexes. The basic connection is often:
1) the functorial correspondence between continuous maps of spaces and homomorphisms of fundamental group, as well as
2) the Galois correspondence between covering spaces and subgroups of the fundamental group.
Unfortunately the relationship between fundamental groups and geometric actions seems to be more difficult when spaces which aren’t locally contractible are allowed. A space is ‘wild’ if it is not homotopy equivalent to a CW complex — a solenoid, or a Menger sponge for instance. I will be talking about some new and quite beautiful facts that help turn the study of ‘wild’ topological spaces from a safari in the land of pathology into an enterprise of taxonomy. Basic interesting open questions to study abound, for instance: “Can a subset of Euclidean 3-space have torsion in its first homology or in its fundamental group”?
There are a few tools that we will talk about:
1) Since not every homomorphism between fundamental groups is realized by a continuous map we need to be able to know when such homomorphisms are representable by good maps. We will talk about slenderness and automatic continuity.
2) Since wild spaces don’t have universal covering spaces we need a replacement for the notion of covering space which has some kind of Galois correspondence. We will talk about lifting spaces, the shape kernel, and path connectivity.
3) If time allows, we will talk about a connection between the notion of slenderness and a long-open question in number theory, the Kurepa Conjecture.