Determine the exact complexity of the membership algorithm

Determine the exact computational complexity of the algorithm deciding Submonoid Membership, and hence Subgroup Membership, in \(\operatorname{FSym}(\mathbb{Z})\rtimes\mathbb{Z}\), including the complexity contribution of the reduction to Subgroup Membership in finite-group wreath products \(H\wr\mathbb{Z}\).

Background

The paper establishes decidability of Submonoid Membership in the lampshuffler group FSym⁡(Z)⋊Z\operatorname{FSym}(\mathbb{Z})\rtimes\mathbb{Z} by reducing the problem to Subgroup Membership in wreath products H≀ZH\wr\mathbb{Z} with finite HH. Although decidability is inherited from the cited result for these wreath products, the paper does not provide a complexity analysis of the resulting algorithm because the exact complexity of the underlying wreath-product Subgroup Membership procedure is itself unresolved.

References

However, we do not know the complexity of our algorithm for Theorem~\ref{thm:submonoid_membership}, as it relies on a reduction to the Subgroup Membership problem in $H \wr \Z$, whose exact complexity remains unknown.

— Subgroup and Submonoid Membership in the lampshuffler of $\mathbb{Z}$  (2610.06108 - Bodart et al., 5 Oct 2026) in Section 1, paragraph “Main result”