Stability and the Morse boundary

Abstract

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.

Quasi‐Mobius Homeomorphisms of Morse boundaries

“…An alternate construction of the Morse boundary is given by the second author and Hume in . Define to be the set of all such that there exists an ‐Morse geodesic in .…”

“…Define to be the set of all such that there exists an ‐Morse geodesic in . By [, Proposition 3.2], we know is ‐hyperbolic in the Gromov 4‐point definition of hyperbolicity. Hence, we may consider its Gromov boundary, , and the associated visual metric .…”

“…(The reader is referred to [, Section 2.2] for a careful treatment of the sequential boundary of a ‐hyperbolic space which is not necessarily geodesic.) It is shown in that there is a natural homeomorphism between and . In particular, is compact.…”

“…An alternate construction of the Morse boundary is given by the second author and Hume in [12]. Define X , and the associated visual metric d (N ) .…”

Searching for Hyperbolicity

“…We view Morse rays as "hyperbolic-like" directions in X. Indeed, it can be shown that these rays share many other nice properties with rays in hyperbolic space [4,10]. For example, if two sides of a triangle are N -Morse, then the triangle is δ-thin where δ depends only on the Morse gauge N .…”