Stability and the Morse boundary

TL;DR

This paper unifies stable subgroups and Morse boundary concepts to analyze hyperbolic features of groups, introduces new invariants, and applies these to various group classes, revealing structural and dimensional properties.

Contribution

It generalizes the relationship between stable subgroups and Morse boundary, introduces stable and Morse capacity dimensions, and applies these to groups like right-angled Artin and mapping class groups.

Findings

Stable subsets of right-angled Artin groups are quasi-isometric to trees.

Stable dimension of mapping class groups is bounded by surface complexity.

Finite stable dimension is inherited in relatively hyperbolic groups.

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 generalize 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 Gromov monster groups have stable dimension at most 2.