Dice Question Streamline Icon: https://streamlinehq.com

Subgroup membership problem in the right-angled Artin group A(C5)

Determine whether the subgroup membership problem is decidable for the right-angled Artin group A(C5), where C5 is the 5-cycle graph.

Information Square Streamline Icon: https://streamlinehq.com

Background

The authors note that B4 embeds A(C5), and that the subgroup membership problem for A(C5) remains unresolved. This instance is of particular interest because A(C5) sits at a boundary where many techniques (e.g., embeddings leading to F2 × F2) do not directly apply, making it a natural test case for understanding subgroup membership in RAAGs and, by extension, in related Artin groups.

References

Related to this, it follows from the proof of Lemma~4.2 that $B_4$ embeds $A(C_5)$, where $C_5$ is a pentagon, and it is also an open problem whether $A(C_5)$ has decidable subgroup membership problem; see Section~5.

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