The infinite dimensional geometry of conjugation invariant generating sets

TL;DR

This paper explores the large-scale geometric structure of groups with specific conjugation-invariant generating sets, revealing infinite-dimensional features and quasi-isometric embeddings of integer lattices.

Contribution

It introduces new geometric insights into groups with conjugation-invariant generating sets, including a novel subsurface projection for the free splitting graph.

Findings

Cayley graphs contain quasi-isometric copies of ^m for all ma0b1a0

Cayley graphs have infinite asymptotic dimension

Recovered known results about the free group edge-splitting graph

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 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 $m\geq 1$.