Applications of Morse Theory to Geometric Group Theory
Abstract: Suppose you slowly immerse a bagel in a (very large) cup of coffee, all the time observing the shape of the top level of the liquid. We begin with a point of contact when the bagel first hits the coffee. This expands into an oval and then the oval deforms into a figure eight shape just as the "hole" of the bagel begins to be immersed. The figure eight splits into two ovals which move apart and eventually reconverge to yield another figure eight just as the "hole" is completely submerged. Then this figure eight breaks into an oval again which finally shrinks to a point as the bagel is completely submerged.
Another way of describing this is to say that the height function on the bagel has a local max, a local min, and two saddle points. Classical Morse theory tells us how to describe the shape of a manifold -- in this case a torus (the surface of the bagel) -- in terms of the critical point data of a height (or Morse) function on the manifold.
In this talk we shall see that the essentials of the Morse theory framework carry over to a large class of spaces which are not manifolds, and we shall describe some of the many interesting applications that emerge from this setting. The basic philosophy behind these applications is the following:
"Given a Morse function on some reasonable space. Then local (critical point) data together with nice global properties of the space give structural information about level sets (horizontal slices). This in turn gives information about groups acting on the space."