  We introduce a new geometric tool for analyzing groups of finite automata. To
each finite automaton we associate a square complex. The square complex is
covered by a product of two trees iff the automaton is bi-reversible. Using
this method we give examples of free groups and of Kazhdan groups which are
generated by the different states of one finite (bi-reversible) automaton. We
also reproduce the theorem of Macedonska, Nekrashevych, Sushchansky, on the
connection between bi-reversible automata and the commensurator of a regular
tree.

Group Theory