Expander families, group structure, and semidirect products

Expander families are, essentially, sequences of large, sparse, pseudorandom graphs. Many such families have been constructed as Cayley graphs. It is an interesting and open question to determine which groups yield Cayley graphs that form expander families. In this paper, we give a brief survey of expander families, with an emphasis on known results pertaining to that question. One minor new result is a necessary but insufficient condition for a sequence of finite solvable groups, each constructed by iterating semidirect products, to yield an expander family as a sequence of Cayley graphs.