On amenability of automata groups
  We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups ("Mother groups") which is done by analyzing the asymptotic properties of random walks on these groups. 
Laurent Bartholdi, Vadim A. Kaimanovich, Volodymyr V. Nekrashevych