Semi-direct product in groups and zig-zag product in graphs: connections and applications

We consider the standard semi-direct product A×B of finite groups A, B. We show that with certain choices of generators for these three groups, the Cayley graph of A×B is (essentially) the zigzag product of the Cayley graphs of A and B. Thus, using the results of O. Reingold et al. (2000), the new Cayley graph is an expander if and only if its two components are. We develop some general ways of using this construction to obtain large constant-degree expanding Cayley graphs from small ones. A. Lubotzky and B. Weiss (1993) asked whether expansion is a group property; namely, is being an expander for (a Cayley graph of) a group G depend solely on G and not on the choice of generators. We use the above construction to answer the question in the negative, by showing an infinite family of groups A_i×B_i which are expanders with one choice of a (constant-size) set of generators and are not with another such choice. It is interesting to note that this problem is still open, though for "natural" families of groups like the symmetric groups S_n or the simple groups PSL(2, p).