I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:

(1) Heisenberg group $\mathbb{H}(\mathbb{Z})$ (result of Cheeger and Kleiner, Ann. of Math. 171 (2010), 1347-1385).

(2) Gromov's random group (also known as Gromov's monster), Geom. Funct. Anal. 13 (2003), 73-146. These groups do not admit even coarse embeddings into $L_1$ since they contain families of expanders weakly; there are further examples of Osajda arXiv:1406.5015 which do not admit bilipschitz embeddings into $L_1$ for the same reason.

Question: Any other known examples?

Remark. Finitely generated Gromov hyperbolic groups admit bilipschitz embeddings into $L_1$ by the result of Buyalo, Dranishnikov, and Schroeder, Invent. Math. 169 (2007), 153-192. The same is true for finite direct products of such groups.