Abstract: Groups tend to be most interesting when they arise as symmetry groups of some geometric object. The groups we will be particularly interested in arise in the study of graphs and surfaces. We will explore how homotopy theoretic methods can be applied to understand their homological properties and natural maps between them. Somewhat counter intuitively, the problem is often simplified when we study the groups not one at a time but in families -- for example, when studying all symmetric groups at the same time.