Decidability at all levels for group-based concatenation hierarchies

Establish the decidability of membership at every level n for concatenation hierarchies whose bases consist of group languages (e.g., the group hierarchy of Margolis–Pin or the modulo hierarchy). Specifically, prove that membership is decidable for all full levels in these hierarchies.

Background

Group-based hierarchies (including the group and modulo hierarchies) are prominent concatenation hierarchies built from bases consisting of group languages or slight extensions. Numerous results exist for low levels (e.g., level one decidability and certain covering/separation results), and this paper derives level-two and level-three decidability in some cases via reductions.

However, a general proof that membership is decidable for all levels across these group-based hierarchies remains unresolved, and is identified as a longstanding open problem.