Obstructions to coarse universality for finitely generated groups

Abstract

We prove that, for every bounded-degree graph $Λ$ admitting a finitely cobounded coarse quasi-action by a group, there is a finitely generated group which does not coarsely embed into $Λ$. More generally, for every countable family $(Λ_i)$ of such graphs, there is a finitely generated group that does not coarsely embed into any $Λ_i$. This resolves two conjectures of Simon Thomas: neither a universal Cayley graph nor a universal quasi-isometry class of finitely generated groups exists. As another consequence, we show that no locally compact second countable group coarsely contains every finitely generated group. The proof uses an exponential upper bound on the number of finite graphs admitting an $(L,M)$-regular map into $Λ$, together with a superexponential supply of high-girth $3$-regular graphs, yielding a sequence of finite high-girth obstruction graphs. A graphical small-cancellation labeling, using a variation of Osajda's labeling theorem following Esperet and Giocanti, then realizes this sequence isometrically inside the Cayley graph of a finitely generated group.