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Stability and the Morse boundary
- Source :
- Journal of the London Mathematical Society. 95:963-988
- Publication Year :
- 2017
- Publisher :
- Wiley, 2017.
-
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.<br />Updated with comments from the referee. To appear in the Journal of the London Mathematical Society. 28 pages, 1 figure
- Subjects :
- Surface (mathematics)
Pure mathematics
General Mathematics
010102 general mathematics
Boundary (topology)
Metric Geometry (math.MG)
Geometric Topology (math.GT)
Group Theory (math.GR)
16. Peace & justice
01 natural sciences
Mapping class group
Mathematics - Geometric Topology
Metric space
Mathematics - Metric Geometry
Dimension (vector space)
Bounded function
0103 physical sciences
FOS: Mathematics
Artin group
20F65, 20F67
010307 mathematical physics
0101 mathematics
Invariant (mathematics)
Mathematics - Group Theory
Mathematics
Subjects
Details
- ISSN :
- 14697750 and 00246107
- Volume :
- 95
- Database :
- OpenAIRE
- Journal :
- Journal of the London Mathematical Society
- Accession number :
- edsair.doi.dedup.....de9721e4456f59b0862e081ae2deb4fd