Decidability of general systems of equations in Baumslag–Solitar groups BS(1,p)

Determine whether the Equation System Problem is decidable in the Baumslag–Solitar group BS(1,p) for p ≥ 2; given finite words w1, …, wt over variables X ∪ X^{-1} and constants from BS(1,p), decide whether there exist elements g1, …, gn ∈ BS(1,p) such that w1(g1, …, gn) = ⋯ = wt(g1, …, gn) = e.

Background

The paper surveys algorithmic results for abelian-by-cyclic groups and provides new undecidability and decidability results. For Baumslag–Solitar groups BS(1,p), prior work has established that solving systems of quadratic equations is decidable and NP-complete, and rational subset membership is PSPACE-complete. However, the entry for general systems of equations in BS(1,p) remains marked as open in the summary table.

The authors emphasize that question marks in the table indicate open problems. In particular, in the row for BS(1,p) under the column ‘General Equations’, the entry is a question mark, signifying that the decidability of general systems of equations in BS(1,p) is unknown.