Morse and stable subgroups via the coset intersection complex

Abstract: In this note, we study the equivalence of Morse and stable subgroups in the framework of the coset intersection complex. Under certain conditions on a coset intersection complex of a group, we prove that infinite-index Morse subgroups are stable. Our main theorem recovers results in the literature on right-angled Artin groups and graph products. As an application, we show that for the genus-two handlebody group, any infinite-index Morse subgroup is stable.

• The paper establishes a unifying framework showing every infinite-index Morse subgroup under specific conditions is stable and virtually free.
• It employs the coset intersection complex to analyze subgroup intersections, leveraging finite height properties and quasi-tree structures.
• The study recovers known results for right-angled Artin groups, graph products, and genus-two handlebody groups, deepening the understanding of subgroup geometry.

Morse and Stable Subgroups via the Coset Intersection Complex

Overview

This paper establishes a unifying framework for understanding the relationship between Morse (strongly quasiconvex) subgroups and stable subgroups in finitely generated groups, utilizing the coset intersection complex as the central tool. The main theorem provides sufficient conditions under which every infinite-index Morse subgroup is stable, further showing such subgroups are virtually free. The approach recovers and generalizes earlier results for right-angled Artin groups, graph products, and the genus-two handlebody group.

Morse and Stable Subgroups: Definitions and Prior Work

The paper adopts the following definitions in the context of finitely generated groups $G$ and subgroups $H$:

• $H$ is stable if it is finitely generated, undistorted, and any pair of $(k, c)$-quasigeodesics with endpoints in $H$ remain uniformly close in $G$.
• $H$ is Morse (strongly quasiconvex) if all $(k, c)$-quasigeodesics with endpoints on $H$ remain in a uniform neighborhood of $H$ in $H$0.

The equivalence of these notions only holds in hyperbolic groups. Prior counterexamples in right-angled Coxeter and peripheral subgroups of relatively hyperbolic groups establish that the equivalence fails in general; however, it does hold in mapping class groups of surfaces of sufficient complexity [3978121], right-angled Artin groups [3956891], and certain graph products [balasubramanya2026stablesubgroupsgraphproducts].

Tran's characterization connects stability, the Morse property, and hyperbolicity: an infinite Morse subgroup is stable if and only if it is hyperbolic [3956891, Theorem 4.8].

The Coset Intersection Complex and Characterization Theorem

The focus of the paper is the coset intersection complex $H$1 associated to a group $H$2 and a finite family of infinite subgroups $H$3. The vertices are left cosets $H$4 ($H$5), and a collection of $H$6 cosets forms an $H$7-simplex if the intersection of their conjugates is infinite.

The main result can be summarized as follows (rewritten for clarity):

Theorem:

Let $H$8 be a group with a finite family $H$9 of infinite subgroups satisfying:

1. Each $H$0 is undistorted in $H$1.
2. Every Morse subgroup of $H$2 is either finite or finite-index in $H$3.
3. $H$4 is quasi-isometric to a simplicial tree.

Then every non-trivial infinite-index Morse subgroup $H$5 of $H$6 is stable and virtually free.

The implications are significant: In the presence of the above structural properties, Morse and stable subgroups coincide in a strong sense. Moreover, the restriction to the setting where the coset intersection complex is quasi-tree-like yields a virtual freeness result for Morse subgroups, connecting geometric group theory with outcomes reminiscent of Stallings' theorem and accessibility [807066, 228573].

Applications to Group Classes

Graph Products:

If $H$7 is a graph product of infinite groups indexed by the vertices of a finite, connected graph with at least two vertices, and $H$8 is the collection of subgroups generated by cliques, then every infinite-index Morse subgroup of $H$9 is stable and virtually free. These conditions are met because (a) subgroup retracts ensure undistortedness, (b) direct products of infinite groups only have finite or finite-index Morse subgroups [balasubramanya2026stablesubgroupsgraphproducts, Proposition 6.4], and (c) the associated coset intersection complex is a quasi-tree [abbott2025homotopytypescomplexeshyperplanes].

Groups Acting on Trees:

For a group acting cocompactly and acylindrically on a simplicial tree (with infinite, undistorted edge stabilizers with only finite or finite-index Morse subgroups), the conditions are similarly met, and the conclusions apply.

The Genus-Two Handlebody Group:

The handlebody group $(k, c)$0 is a subgroup of the genus-$(k, c)$1 mapping class group and acts on the disk complex. The paper shows that in $(k, c)$2, any infinite-index Morse subgroup is stable and virtually free. The proof leverages (a) undistortedness of relevant Dehn twist subgroups, (b) the structure of such subgroups as being either finite or finite-index, and (c) quasi-isometry of the coset intersection complex to a simplicial tree, via the disk complex [4464468, 4235016].

Technical Contributions

The technical heart of the paper is the use of finite height properties of Morse subgroups and a careful analysis of intersections with conjugates of peripheral-like subgroups. Key ingredients include:

• Preservation of Morse and undistorted properties under conjugation.
• Reduction to proper action on quasi-trees, yielding virtual freeness via Stallings' theorem and Dunwoody's accessibility.
• A lifting argument in the coset intersection complex, guaranteeing that large intersection of a Morse subgroup with peripheral-like conjugates is precluded unless the subgroup is not infinite-index.

The framework encompasses and streamlines proofs from the mapping class group, right-angled Artin group, and graph product literature, demonstrating the efficacy of the coset intersection complex as an organizing tool.

Open Problems and Outlook

The paper raises the question whether these results hold whenever the coset intersection complex is hyperbolic, not just quasi-tree. Positive indications exist for mapping class groups (via quasi-isometric embeddings into the hyperbolic curve complex) but the general case remains open. A further challenge is extending the results to higher-genus handlebody groups, where the associated coset intersection complexes are hyperbolic but not quasi-tree [2983005, 4464468].

From a practical viewpoint, this research indicates that detecting Morse or stable subgroups in large classes of groups (notably, those with graph-of-groups structures or suitable tree actions) can be reduced to the structure of the coset intersection complex and the behavior of peripheral subgroup Morse geometry. Theoretical implications include the rigidity of Morse subgroups under the stated conditions and the deep interplay between group action geometry and subgroup algebraic properties.

Conclusion

This work provides a cohesive, general theorem that clarifies when Morse and stable subgroups in a wide range of groups coincide and exhibit virtual freeness, unified under the coset intersection complex framework. The methods not only recover known results but also outline a clear path for further exploration in both the structure theory of groups and the geometry of subgroups. Extensions to broader settings, especially coset intersection complexes of hyperbolic type, represent compelling directions for future work.

Reference:

"Morse and stable subgroups via the coset intersection complex" (2603.29158)