Automata and Square complexes
(With Shahar Mozes)
Abstract.
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.