$L_1$ and $L_2$ embeddings of the symmetric group

TL;DR

This paper demonstrates that certain Cayley graphs of the symmetric group can be embedded into L1 space with low distortion, revealing new insights into group embeddings and the Ribe program.

Contribution

It provides the first example of groups with different bi-Lipschitz embeddability into L1 depending on the generating set, advancing understanding of geometric group theory.

Findings

Cayley graphs generated by specific permutations embed into L1 with O(1) distortion.

Different generating sets can alter the embeddability into L1.

Cayley graphs in this context do not contain coarsely unbounded expanders.

Abstract

We show that the Cayley graph of the symmetric group $Sym_n$ generated by the cycle $(123...n)$ and the transposition $(12)$ embeds into $L_1$ with bi-Lipschitz distortion $O(1)$. This answers a question of Ostrovskii, and along with Kassabov's theorem gives the first example of a sequence of groups which embed bi-Lipschitzly into $L_1$ for one choice of bounded size generating sets, but not for another choice of bounded size generating sets. In particular, the Cayley graphs generated by the cycle and the transposition cannot contain coarsely any unbounded sequence of expander graphs. Moreover, within the context of the Ribe program, they are a new example of bounded degree Cayley graphs which are test spaces for Rademacher type.