We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper CAT(0) space this boundary is the contracting boundary of Charney and Sultan, and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichmüller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichmüller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichmüller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichmüller space into the Thurston compactification of Teichmüller space by projective measured foliations.
Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalise these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a quasi-convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces.We use this approach, together with results of Leininger-Schleimer, to deduce that there is no purely geometric obstruction to the existence of a non-virtually-free convex cocompact subgroup of a mapping class group.In addition, we define two new quasi-isometry invariant notions of dimension: the stable dimension, which measures the maximal asymptotic dimension of a stable subset; and the Morse capacity dimension, which naturally generalises Buyalo's capacity dimension for boundaries of hyperbolic spaces.We prove that every stable subset of a right-angled Artin group is quasi-isometric to a tree; and that the stable dimension of a mapping class group is bounded from above by a multiple of the complexity of the surface. In the case of relatively hyperbolic groups we show that finite stable dimension is inherited from peripheral subgroups.Finally, we show that all classical small cancellation groups and certain graphical small cancellation groups -including some Gromov monster groups -have stable dimension at most 2.
A Kleinian group Γ < Isom(H 3 ) is called convex cocompact if any orbit of Γ in H 3 is quasiconvex or, equivalently, Γ acts cocompactly on the convex hull of its limit set in ∂H 3 .Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups.Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.
The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic‐like behavior in the space. A key property of this boundary is that a quasi‐isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper, we investigate when the converse holds. We prove that for X,Y proper, cocompact spaces, a homeomorphism between their Morse boundaries is induced by a quasi‐isometry if and only if the homeomorphism is quasi‐mobius and 2‐stable.
We study direct limits of embedded Cantor sets and embedded Sierpiński curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called !-Cantor spaces, and, similarly, all limits of Sierpiński curves give homeomorphic spaces, called !-Sierpiński curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups and fundamental groups of non-geometric graph manifolds, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds.
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