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Rational subset membership problem for hyperbolic surface groups

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

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Background

In discussing potential classifications for Coxeter groups with decidable membership problems, the authors point to hyperbolic surface groups as a key obstacle: many Coxeter groups are virtually such groups. The status of rational subset membership in this class directly impacts broader classification efforts in Coxeter groups.

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