Automata groups generated by Cayley machines of groups of nilpotency class two

We build presentations for automata groups generated by Cayley machines of finite groups of nilpotency class two and prove that these automata groups are all cross-wired lamplighter groups.

Non-elementary amenable subgroups of automata groups

We consider groups of automorphisms of rooted locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. We give a unified proof for all known examples of non-elementary amenable groups that act on the trees: groups of intermediate growths and Basilica group. Moreover, we show that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.

Groups and semigroups defined by colorings of synchronizing automata

In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the de Bruijn automata into Mealy automata whose associated semigroups are free, and we present some structural results related to the associated groups.