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Decidability of general systems of equations in lamplighter groups Z_p wr Z

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

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Background

Within the broader class of abelian-by-cyclic groups, the lamplighter groups Z_p wr Z have seen significant progress on specific problems (e.g., NP-completeness for quadratic equations and decidability of rational subset membership).

Despite these advances, the table summarizing the state of the art marks the entry for general systems of equations in Z_p wr Z with a question mark, indicating that the decidability of solving general systems of equations in Z_p wr Z remains an open problem.

References

The following table summarizes our results for abelian-by-cyclic groups in the context of the current state of art. The question marks denote open problems. Results for empty blocks are subsumed by results in the same row or column.

— Linear equations with monomial constraints and decision problems in abelian-by-cyclic groups (2406.08480 - Dong, 12 Jun 2024) in Table 1, Section 1 (Introduction and main results)