Stability of infinite-index Morse subgroups when the coset intersection complex is hyperbolic

Determine whether the following holds: For any finitely generated group pair (G, P) such that the coset intersection complex K(G, P) is hyperbolic, each subgroup P in the finite collection P is undistorted in G, and every Morse subgroup of each P is either finite or finite-index in P, every infinite-index Morse subgroup H ≤ G is stable.

Background

The paper investigates when Morse (strongly quasiconvex) subgroups coincide with stable subgroups. The main theorem establishes that, under three conditions—including that the coset intersection complex K(G, P) is quasi-isometric to a tree—every infinite-index Morse subgroup is stable and virtually free. This recovers known results for right-angled Artin groups and certain graph products.

However, this quasi-tree hypothesis is stronger than hyperbolicity. The authors note they cannot recover the mapping class group case via their main theorem and therefore pose a broader question: does the conclusion remain true assuming only that K(G, P) is hyperbolic while retaining the other hypotheses (undistorted P and that Morse subgroups of P are finite or finite-index)? They provide positive evidence from mapping class groups, where K(Mod(S), P) is quasi-isometric to the curve complex (known to be hyperbolic), and the other hypotheses hold.