The infinite dimensional geometry of conjugation invariant generating sets

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 work of Brandenbursky-Gal-K\c{e}dra-Marcinkowski, in these Cayley graphs we show that there are quasi-isometrically embedded copies of $\mathbb{Z}^m$ for all $m\geq 1$. 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 $m\geq 1$.