Relatively hyperbolic groups with fixed peripherals

“…Variations of the constructions in this paper are ideally suited to the study of relatively hyperbolic groups. These constructions will be the focus of a future paper ; the main result of which states that given any finite collection of non‐relatively hyperbolic groups

scriptH

, there are infinitely many 1‐ended groups which are hyperbolic relative to

scriptH

. In the specific case where each

H \in H

has finite stable dimension, one can find a family of such relatively hyperbolic groups with unbounded stable dimension.…”

Section: Introductionmentioning

confidence: 99%

Stability and the Morse boundary

Cordes

Hume

2017

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In [CH16] Cordes-Hume focus on relatively hyperbolic groups. In this paper they suggest an approach to answering the following question which appears in [BDM09]: How may we distinguish non-quasi-isometric relatively hyperbolic groups with non-relatively hyperbolic peripheral subgroups when their peripheral subgroups are quasi-isometric?…”

Section: Relatively Hyperbolic Groupsmentioning

confidence: 99%

A survey on Morse boundaries & stability

Cordes¹

2017

Preprint

Self Cite

View full text Add to dashboard Cite

Recently there have been efforts to generalize tools used in the setting of Gromov hyperbolic spaces to larger classes of spaces. We survey here two aspects of these efforts: Morse boundaries and stable subspaces. Both the Morse boundary and stable subspaces are systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. We will see in this survey a nice relationship between the two approaches.Throughout the survey we will expect the reader to be familiar with the basics of geometric group theory. For details on this material see [BH99] and for additional information especially on boundaries of not-necessarily-geodesic hyperbolic spaces see [BS07].We begin with an important definition that has its roots in a classical paper of Morse [Mor24]:

show abstract

Relatively hyperbolic groups with fixed peripherals

Cited by 3 publications

References 33 publications

Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups

Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups

Stability and the Morse boundary

A survey on Morse boundaries & stability

Contact Info

Product

Resources

About