We consider the standard semi-direct product $A\rtimes B$ of
finite groups $A,B$. We show that with certain choices of
generators for these three groups, the Cayley graph of
$A\rtimes B$ is (essentially) the zigzag product of the
Cayley graphs of $A$ and $B$. Thus, using the results of
\cite{RVW}, 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 {\em constant-degree}
expanding Cayley
graphs from small ones.
In \cite{LW}, 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 groups
$A_i\rtimes B_i$ 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 $S_n$ or the simple
groups $PSL(2,p)$.