  The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We show
that the set of occupied sites for this model on an infinite regular tree is a
perfect ball whenever it can be, provided the initial rotor configuration is
acyclic (that is, no two neighboring vertices have rotors pointing to one
another). This is proved by defining the rotor-router group of a graph, which
we show is isomorphic to the sandpile group. We also address the question of
recurrence and transience: We give two rotor configurations on the infinite
ternary tree, one for which chips exactly alternate escaping to infinity with
returning to the origin, and one for which every chip returns to the origin.
Further, we characterize the possible "escape sequences" for the ternary tree,
that is, binary words a_1 ... a_n for which there exists a rotor configuration
so that the k-th chip escapes to infinity if and only if a_k=1.
