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Submonoid membership problem for hyperbolic surface groups

Determine whether the submonoid membership problem is decidable for hyperbolic surface groups.

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Background

The authors identify hyperbolic surface groups as central to future classifications of Coxeter groups regarding membership problems. Since many Coxeter groups are virtually hyperbolic surface groups, the unresolved status of submonoid membership in this class presents a critical barrier.

References

One current obstacle to this is that there are many Coxeter groups (even right-angled Coxeter groups) that are virtually hyperbolic surface groups (see e.g. Theorem~2.1), and the rational and submonoid membership problems are both currently open for hyperbolic surface groups.

— Membership problems in braid groups and Artin groups (2409.11335 - Gray et al., 17 Sep 2024) in Section 4 (Artin groups), final paragraph