ar X iv : m at h / 04 12 37 3 v 2 [ m at h . G R ] 2 1 O ct 2 00 5 AUTOMATA , GROUPS , LIMIT SPACES , AND TILINGS

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.