The local-to-global property for Morse quasi-geodesics

Abstract

We show the mapping class group, CATp0q groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasigeodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

Extensions of multicurve stabilizers are hierarchically hyperbolic

“…The main motivation is to show that certain hierarchically hyperbolic groups will also have ω-Cantor space boundaries. However, as the full power of hierarchical hyperbolicity will not be needed for our proof, we will instead work in the simpler setting of Morse detectable groups introduced by the author, Spriano, and Tran [RST19].…”

Regularity of Morse geodesics and growth of stable subgroups

“…To apply Proposition 5.2 to the situations in Theorem 5.1, we employ the following results of Russell, Spriano, and Tran to find the required H ′ and g ∈ G. Both of these results are consequence of a general combination theorem for stable subgroups of Morse local-to-global groups [RST19].…”

“…Gromov showed that hyperbolic spaces are characterized by local quasi-geodesics being global quasi-geodesics [Gro87]; the Morse local-toglobal property is a generalization of this phenomena. Morse local-to-global groups were introduced by Russell, Spriano, and Tran who showed they include the mapping class group of an orientable, finite type surface, all cocompact CAT(0) groups, the fundamental group of any closed 3-manifold, and any group hyperbolic relative to Morse local-to-global groups [RST19]. We will describe below the two places in the proof of Theorem A where the Morse local-to-global property is used.…”

“…Given an infinite index stable subgroup H, we use either the virtual freeness of G or the residually finiteness of H to produce an element g ∈ G and a finite index subgroups H ′ < H so that H ′ , g ∼ = H ′ * g . In both cases, this relies on combination theorems for Morse local-to-global groups proved by Russell, Spriano, and Tran [RST19]. Since finite index subgroups of stable subgroups are stable, we can apply Theorem E to H ′ to produce a regular language J H ′ of geodesic words that uniquely represent the elements of H ′ .…”

Random walks and quasi-convexity in acylindrically hyperbolic groups

“…If such a subgroup exists, then Corollary 1.8 allows one to construct more examples of non-free convex cocompact subgroups. Note that another way to combine convex cocompact subgroups into potentially new convex cocompact subgroups also appeared recently in [57].…”