Negative curvature in graphical small cancellation groups

Abstract

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical Gr ( 1 /6) small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically.We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group G containing an element g that is strongly contracting with respect to one finite generating set of G and not strongly contracting with respect to another. In the case of classical C ( 1 /6) small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting.We show that many graphical Gr ( 1 /6) small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups.In the course of our analysis we show that if the defining graph of a graphical Gr ( 1 /6) small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.

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

“…We thank David Hume for pointing out to us that not all finitely generated groups have the stable coherence property; for example, see [, Theorem 6.6]. However, there are no known counterexamples when is finitely presented.…”

Relative Hyperbolicity of Graphical Small Cancellation Groups

“…This suggested an analogy with relatively hyperbolic space: the embedded components of the defining graph corresponding to the peripheral regions of relatively hyperbolic space. This is also well informed in the work of Arzhantseva-Cashen-Gruber-Hume [1]. Informally, typical results there say that geodesics which have bounded penetration into embedded components are strongly contracting, hence they behave like hyperbolic geodesics.…”

“…First of all, we follow the approach of [1] by using Strebel's classification of combinatorial geodesic triangles to show that A is a strongly contracting system. Thus, the properties (Λ 1 ) and (Λ 2 ) of A follow from the strongly contracting property.…”

“…Note that any simple bigon with edges limit geodesics can be represented as an ω-limit of a sequence 'fat' simple geodesic quadrangles by the strategy in the proof of [4,Proposition 4.14]. However, the classification of 'special' combinatorial quadrangles provided by Arzhantseva-Cashen-Gruber-Hume [1] shows us that there is no 'fat' special combinatorial quadrangle. This gives us great help to show the property (Ω 2 ).…”

Cogrowth for group actions with strongly contracting elements

“…Results (3)-(5) are new, only known as consequences of Theorem 1.1. Further new examples include wide classes of snowflake groups [2] and of infinitely presented graphical and classical small cancellation groups [1], hence, many so-called infinite 'monster' groups.…”