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

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.