Decision Problems for Automaton Groups and Monoids of Bounded ActivityTBA

the class of finite groups and the next level is the class of automaton groups with bounded activity (often simply called bounded automaton groups). This level already contains many of the famous automaton groups (in fact, it contains the examples mentioned above). Surprisingly, however, this level of the hierarchy somehow still seems to be “finite enough” for most algebraic decision problems to be decidable (which is not the case for general automaton groups where decision problems are usually at least suspected to be undecidable).

In the talk, we will look at some recent results in this context with an emphasis on the membership problem for monogenic subsemigroups (which was proven to be decidable by Bühler in his Master thesis) and the finiteness
problem (where we will discuss some ongoing research results on its decidability). Both of these decidability results are roughly based on constructing a finite weighted automaton computing orbit sizes. With regard to future research, the nature of this automaton promises the potential to be useful for further algebraic decision problems in this area and we will also see an outlook on this.