Small Cancellation Labellings of Some Infinite Graphs and Applications

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely nonamenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs – in contrast, for the Gromov monster only a weak embedding is established. Such examples have important applications to the Baum-Connes conjecture. We present further applications concerning aspherical manifolds and asymptotic dimension of groups.