Embedding and commensurability problems for right-angled Artin groups

Determine whether right-angled Artin groups embed into one another as subgroups and classify right-angled Artin groups up to commensurability.

Background

The introduction identifies two longstanding problems for right-angled Artin groups beyond the quasi-isometry classification problem: determining when one right-angled Artin group embeds into another and classifying right-angled Artin groups up to commensurability. These problems are explicitly described as unresolved, although the paper focuses on quasi-isometric invariants rather than solving them.

References

Despite the fact that right-angled Artin groups have been extensively studied, a few very natural questions remain open, including the embedding problem, the classification up to commensurability, and, from the perspective of geometric group theory, the following question to which our article is dedicated:

— Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups  (2503.08411 - Abbott et al., 11 Mar 2025) in Section 1, Introduction