In the 1960's, Richard J. Thompson described three groups F, T, and V, which act by homeomorphisms on the interval, the circle, and the Cantor set, respectively.  In this talk, I will describe an analogous group that acts by homeomorphisms on the Basilica Julia set.  This group can also be described as a group of piecewise-linear homeomorphisms of the unit circle that preserves the invariant lamination determined by the Basilica.  I will sketch a proof that this group is finitely generated and virtually simple, and discuss possible generalizations to other Julia sets.  This is joint work with Bradley Forrest.