Kazhdan's property (T) and expander graphs
An expander graph is a well-connected finite graph, with one consequence being that random walks mix extremely quickly on them. While it is relatively easy to show that they exist, and in some sense most graphs are expanders, constructing explicit examples is non-trivial. Margulis gave the first construction using Kazhdan's property (T), a representation-theoretic property of groups. I'll explain this construction, and possibly further developments.