Semi-Direct Product in Groups and Zig-Zag Product in Graphs: Connections and Applications

We consider the standard semi-direct product AoB of finite groups A,B. We show that with certain choices of generators for these three groups, the Cayley graph of A o B is (essentially) the zigzag product of the Cayley graphs of A and B. Thus, using the results of [RVW00], 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. In [LW93], Lubotzky and Weiss asked whether expansion is a group property; namely, is being 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 negative, by showing an infinite family of groupsAioBi which are expanders with one choice of (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 Sn or the simple groups PSL(2, p). ∗Schools of Mathematics and Computer Science, Tel-Aviv University, Tel Aviv, E-mail: [email protected]. Partially supported by a US-Israel BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. †Department of Mathematics, The Hebrew University, Jerusalem, E-mail: [email protected]. Partially supported by NSF and US-Israel BSF grants. ‡Institute for Advanced Study, Princeton and the Hebrew University, Jerusalem. E-mail: [email protected]. Partially supported by NSF grants CCR-9987007 and CCR-9987845