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 ${\mathrm{Gr}}^{\prime }(1/6)$ 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 $C^{\prime }(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting.

We show that many graphical ${\mathrm{Gr}}^{\prime }(1/6)$ 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 ${\mathrm{Gr}}^{\prime }(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.