  We construct rearrangement groups for edge replacement systems, an infinite
class of groups that generalize Richard Thompson's groups F, T, and V .
Rearrangement groups act by piecewise-defined homeomorphisms on many
self-similar topological spaces, among them the Vicsek fractal and many Julia
sets. We show that every rearrangement group acts properly on a locally finite
CAT(0) cubical complex, and we use this action to prove that certain
rearrangement groups are of type F infinity.
