Subgroup and submonoid membership in the remaining Houghton groups

Determine the decidability of the Subgroup Membership and Submonoid Membership problems for the Houghton groups beyond the cases established for the finitary symmetric group and the lampshuffler group of \(\mathbb{Z}\).

Background

The paper identifies H1=FSym⁡(N)H_1=\operatorname{FSym}(\mathbb{N}) and H2=FSym⁡(Z)⋊ZH_2=\operatorname{FSym}(\mathbb{Z})\rtimes\mathbb{Z} as the first two Houghton groups. It notes that the Conjugacy Problem is decidable in all Houghton groups, whereas the corresponding membership problems have not been resolved in the general family. The paper proves decidability for the lampshuffler group H2H_2, leaving the broader question for the remaining Houghton groups unresolved.

References

In particular, the Conjugacy Problem was shown to be decidable in all Houghton groups, while decidability of Subgroup and Submonoid Membership remains open.

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