**Seminario Teoría de Grupos ICMAT-UAM**

**Ponente: Greg Conner (Brigham Young University) **

** Title:‘Some Notes on Wild Geometric Group Theory’. **

**Miercoles 17/10/2018, 11:30, Aula 520, UAM **

** Abstract: **

** 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.**