Polishness of some topologies related to automata (Extended version)

The knapsack problem is a classic optimisation problem that has been recently extended in the setting of groups. Its study reveals to be interesting since it provides many different behaviours, depending on the considered class of groups. In this paper we deal with groups generated by Mealy automata—a class that is often used to study group-theoretical conjectures—and prove that the knapsack problem is undecidable for this class. In a second time, we construct a graph that, if finite, provides a solution to the knapsack problem. We deduce that the knapsack problem is decidable for the so-called bounded automaton groups, a class where the order and conjugacy problems are already known to be decidable.