Acta Mathematica

Volume 225 (2020)

Number 1

Small cancellation labellings of some infinite graphs and applications

Pages: 159 – 191



Damian Osajda (Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland; and Fakultät für Mathematik, Universität Wien, Austria)


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 non-amenable 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, in the case of the Gromov monster expanders are not even coarsely embedded.

We present further applications.


small cancellation, coarse embedding, Property A, CAT(0) cubical complex, graph coloring

2010 Mathematics Subject Classification

05C15, 20F06, 20F69, 46B85

Received 3 February 2016

Received revised 8 September 2019

Accepted 13 May 2020

Published 4 November 2020