  We explore the connections between automata, groups, limit spaces of
self-similar actions, and tilings. In particular, we show how a group acting
``nicely'' on a tree gives rise to a self-covering of a topological groupoid,
and how the group can be reconstructed from the groupoid and its covering. The
connection is via finite-state automata. These define decomposition rules, or
self-similar tilings, on leaves of the solenoid associated with the covering.
