  The Internal Diffusion Limited Aggregation has been introduced by Diaconis
and Fulton in 1991. It is a growth model defined on an infinite set and
associated to a Markov chain on this set. We focus here on sets which are
finitely generated groups with exponential growth. We prove a shape theorem for
the Internal DLA on such groups associated to symmetric random walks. For that
purpose, we introduce a new distance associated to the Green function, which
happens to have some interesting properties. In the case of homogeneous trees,
we also get the right order for the fluctuations of that model around its
limiting shape.
