Optimization in graphical small cancellation theory (Q6197706)

Let \(\mathcal{G}=(G_{n})_{n \geq 1}\) be a sequence of bounded degree graphs, whose girth tends to infinity. The sequence \(\mathcal{G}\) is \(\mathsf{dg}\)-bounded if the ratio between the diameter and the girth of each \(G_{n}\) is bounded by a constant. \textit{M. Gromov} in [Geom. Funct. Anal. 13, No. 1, 73--146 (2003; Zbl 1122.20021)] proved that there is a finitely generated group \(\Gamma\) whose Cayley graph contains (in a certain metric sense) all the members of \(\mathcal{G}\). By choosing \(\mathcal{G}\) as a family of suitable expander graphs, this implies that such a group \(\Gamma\) has a number of pathological properties, in partucular \(\Gamma\) do not coarsely embed into an Hilbert space. If graphs in Gromov's construction admit graphical small cancellation labellings, then one gets similar examples of Cayley graphs containing all the graphs of the family as isometric subgraphs. \textit{D. Osajda} in [Acta Math. 225, No. 1, 159--191 (2020; Zbl 1512.05352)] showed how to obtain such labellings using the probabilistic method. In the paper under review the authors present a simplified version of the proof of existence of the labelling of Osajda, and significantly decrease the number of generators of \(\Gamma\), and thus the degree of the corresponding Cayley graph. As a concrete example they consider the \(\mathsf{dg}\)-bounded sequence \(\mathcal{G}=(G_{n})\) of cubic Ramanujan graphs introduced by \textit{P. Chiu} in [Combinatorica 12, No. 3, 275--285 (1992; Zbl 0770.05062)] and prove the existence of a group with 96 generators, whose Cayley graph contains all the graphs from \(\mathcal{G}\) as isometric subgraphs. For the same family, the construction of Osajda requires about \(10^{272}\) generators.