Adian–Rabin families of balanced presentations

Determine whether there exists an Adian–Rabin family consisting entirely of balanced finite group presentations.

Background

An Adian–Rabin family is a recursively enumerable family of finite group presentations for which no algorithm can decide whether the group presented by an input presentation is trivial. The paper discusses quantitative restrictions on such families, including fixed numbers of generators and relators and fixed deficiency.

Existing constructions cited in the paper achieve nontrivial bounds on deficiency and relator count, but the stronger requirement that every presentation in the family be balanced remains unresolved. Here, balanced means that the number of generators equals the number of relators.