Stable subgroups of graph products

We extend the characterization of stable subgroups of right-angled Artin groups of Koberda, Mangahas and Taylor to the case of graph products of infinite groups. Specifically, we show that the stable subgroups of such graph products are exactly the subgroups that quasi-isometrically embed in the associated contact graph. Equivalently, they are the subgroups that satisfy a condition arising from the defining graph: a stable subgroup is an almost join-free subgroup. In particular, we generalize the equivalence between stable and purely loxodromic subgroups from Koberda, Mangahas and Taylor in the case where all torsion subgroups of the vertex groups are finite, and the equivalence between stable and infinite index Morse subgroups from Tran in the case where the defining graph is connected.

💡 Research Summary

The paper extends the geometric characterization of stable subgroups from right‑angled Artin groups (RAAGs) to the much broader class of graph products of infinite, finitely generated groups. A graph product Γ G is defined by a finite simplicial graph Γ together with a collection of infinite vertex groups {G_v}. Elements of adjacent vertex groups commute, so the resulting group interpolates between free products and direct products, encompassing RAAGs (when each G_v ≅ ℤ) and right‑angled Coxeter groups (when each G_v ≅ ℤ₂).

The authors first recall the coarse‑geometric notions of quasi‑isometry, stability (undistorted inclusion plus the fellow‑travelling property for quasi‑geodesics), and the various sub‑structures of a graph product such as join, star, and link subgroups. They introduce the key combinatorial condition “almost join‑free”: a subgroup H has only finite intersections with any conjugate of a join subgroup. When Γ has no isolated vertices, “almost join‑free” coincides with “almost star‑free”.

Theorem 1.2 is the central result. For a finitely generated subgroup H ≤ Γ G the following are equivalent:

1. H is stable in Γ G.
2. The orbit map H → C(Γ G) (the contact graph of the prism complex) is a quasi‑isometric embedding.
3. H is almost join‑free.

The contact graph C(Γ G) is known to be a quasi‑tree, hence a Gromov‑hyperbolic space. Consequently it serves as a universal recognizing space for Γ G: every stable subgroup embeds quasi‑isometrically via its orbit map, and any subgroup whose orbit map is a quasi‑isometric embedding must be stable.

The proof relies heavily on disk‑diagram techniques. The authors develop a “join‑busting” analysis that shows almost join‑free subgroups avoid large pieces of join subcomplexes, which forces their images in the contact graph to be quasi‑convex and thus yields a quasi‑isometric embedding. Conversely, stability forces the subgroup to intersect any conjugate of a join subgroup only in a finite set, because otherwise one could construct a pair of quasi‑geodesics with the same endpoints that diverge arbitrarily far, contradicting the fellow‑travelling condition.

When the vertex groups contain no infinite torsion (i.e., every finite subgroup is bounded), the almost join‑free condition simplifies to the purely loxodromic condition. Theorem 1.4 states that under this torsion‑free hypothesis, H is stable if and only if it acts purely loxodromically on the contact graph. This recovers the KMT result for RAAGs and shows that the presence of infinite torsion can break the equivalence (purely loxodromic subgroups need not be stable).

For connected defining graphs, the authors prove Theorem 1.5: a subgroup H ≤ Γ G is stable precisely when it is of infinite index and Morse. This mirrors Tran’s theorem for RAAGs and mapping class groups, but the authors note that the equivalence fails for right‑angled Coxeter groups. The proof uses the fact that in a connected graph product the contact graph is the maximal acylindrical action, and Morse subgroups are exactly those whose orbit maps are quasi‑isometric embeddings.

An immediate corollary (Corollary 1.7) follows from the quasi‑tree nature of the contact graph: any group that quasi‑isometrically embeds into a quasi‑tree is virtually free. Hence all stable subgroups of a graph product of infinite groups are virtually free, extending the known result for RAAGs.

The paper also discusses edge cases. If Γ has isolated vertices, the associated vertex groups are automatically almost join‑free but have finite diameter orbits in the contact graph, so they are stable only when they are hyperbolic. The authors point out that the largest acylindrical action of a graph product need not be a universal recognizing space when isolated vertices are present, referencing recent work of BCK.

Finally, the authors pose Question 1.6: does an analogue of Theorem 1.2 hold when the vertex groups are allowed to be finite? This remains open and suggests further exploration of stability in more general graph products, possibly involving relatively hyperbolic techniques.

Overall, the paper provides a comprehensive framework linking stability, quasi‑isometric embeddings into the contact graph, and combinatorial “almost join‑free” conditions for graph products. It unifies several previously known results for RAAGs, mapping class groups, and Morse subgroups, and establishes the contact graph as the natural hyperbolic model for studying stable subgroups in this broad class of groups.