K(G,1) spaces and Graphs of Groups

In this talk I will introduce K(G,1) spaces, a class of spaces whose homotopy type depends only on their fundamental group.  I will illustrate how these spaces can be used in group theory to give a method for building up a larger group from a collection of smaller groups, generalizing the notion of free products.

To this end, I will introduce the notion of "Graph of Groups", an object consisting of a collection of groups indexed by the vertices and edges of a graph, and ''fundamental group of the graph of groups'' to examine “HNN extensions”, an important tool in combinatorial group theory.