Finitely generated groups non-embeddable into $L_1(0,1)$

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.

• There is not one Gromov random group, there is a recipe providing plenty of groups (and moreover it sometimes means those finitely presented groups containing these infinitely presented groups when they are arranged to be recursively presentable). – YCor Aug 21 '14 at 9:52
• To answer your question, it would be a good PhD work to prove that any non-virtually-abelian nilpotent f.g. group (or, more naturally, any non-abelian simply connected nilpotent Lie group) has no bilipschitz embedding into $L^1$. – YCor Aug 21 '14 at 9:54
• What other classes of f.g. groups do Lipschitz embed into $L_1$? – Bill Johnson Aug 21 '14 at 15:04
• @Bill Free products of groups mentioned in the remark. – Mikhail Ostrovskii Aug 21 '14 at 15:21
• @YCor (Yves) Thank you for the comments. – Mikhail Ostrovskii Aug 21 '14 at 15:21