The infinite-dimensional geometry of conjugation-invariant generating sets

Abstract

We consider a number of examples of groups together with an infinite conjugation-invariant generating set, including the free group with the generating set of all separable elements, surface groups with the generating set of all non-filling curves, mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements, and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. Building on the work of Brandenbursky–Gal–Kędra–Marcinkowski, in these Cayley graphs, we show that there are quasi-isometrically embedded copies of ${\mathbb{Z}}^m$ for all $m\ge 1$. A corollary is that these Cayley graphs have infinite asymptotic dimension. By additionally building a new subsurface projection analogue for the free-splitting graph, which is valued in the above Cayley graph of the free group and may be of independent interest, we are able to recover Sabalka–Savchuk’s result that the edge-splitting graph of the free group has quasi-isometrically embedded copies of ${\mathbb{Z}}^m$ for all $m\ge 1$.