galois groups with finite ramification over $q$ are the "fundamental groups" of number theory. most of what we know about them stems from their action on certain $p$-adic vector spaces. in this talk, i will describe their action on certain trees, which promises to throw a different kind of light on fundamental groups. let $k$ be a number field and $f(x)$ in $k[x]$ be a polynomial whose critical points are preperiodic under iteration of $f$. then every $k$-rational specialization of the tower of iterates of $f:p^1 --> p^1$ is finitely ramified. this leads to a number of open problems about the nature of the corresponding "iterated monodromy" representations of the galois group of $k$. this is joint work with christian maire (toulouse) and wayne aitken (csu san marcos).