Decidability of the freeness problem for automaton groups

Determine whether the freeness problem for automaton groups is decidable; namely, given a complete invertible letter-to-letter transducer (a G-automaton), decide whether the generated automaton group is a free group.

Background

The authors prove that the freeness problems for automaton semigroups and monoids are undecidable, leaving the group case unresolved.

Clarifying the decidability of freeness for automaton groups would be a major step in the algorithmic understanding of automaton groups, complementing recent progress on order, conjugacy, and word problems.