We prove that infinitely presented graphical Gr(7) small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical C(7)-groups and, hence, classical C ( 1 6 )-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical C ( 1 6 )-groups that provide new examples of divergence functions of groups.

We extend fundamental results of small cancellation theory to groups whose presentations satisfy the generalizations of the classical C(6) and C(7) conditions in graphical small cancellation theory. Using these graphical small cancellation conditions, we construct lacunary hyperbolic groups and groups that coarsely contain prescribed infinite sequences of finite graphs. We prove that groups given by (possibly infinite) graphical C(7) presentations contain non-abelian free subgroups.

Abstract. We prove that infinitely presented classical C(6) small cancellation groups are SQ-universal. We extend the result to graphical Gr * (6)-groups over free products. For every p ∈ N, we construct uncountably many pairwise nonquasi-isometric groups that admit classical C(p)-presentations but no graphical Gr ( 1 6 )-presentations.

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.

We show that Gromov’s monster groups arising from i.i.d. labelings of expander graphs do not admit non-elementary actions on geodesic hyperbolic spaces. The proof relies on comparing properties of random walks on randomly labeled graphs and on groups acting non-elementarily on hyperbolic spaces.