Embeddings of finitely generated groups into uniformly convex Banach spaces

Question

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if and only if $G$ contains a subgroup of finite index isomorphic to $\mathbb{Z}^n$ for some $n$. As far as I know this conjecture is still open. I would like to know whether the situation changes if we allow any uniformly convex space as a target space. More precisely:

Question: Does there exist an infinite finitely generated group $G$ such that

(1) $G$ does not contain a subgroup of finite index isomorphic to $\mathbb{Z}^n$; (2) $G$, endowed with its word metric, admits a bilipschitz embedding into some uniformly convex Banach space?

Answer 1

A negative answer to (1) sounds like a natural extension of our conjecture.

Some evidence: in the main two cases for which the conjecture is known to hold in the Hilbert case, the same argument also works for arbitrary uniformly convex Banach spaces (UCB): on the one hand non-virtually abelian nilpotent groups (which don't embed bilipschitz into a UCB, by a result of Pauls), and groups with a bilipschitz-embedded 3-regular tree (which don't embed into a UCB by Bourgain, and include non-amenable f.g. groups by Benjamini-Schramm, and also include non-virtually nilpotent solvable f.g. groups).