Is it known whether the Brin-Thompson 2V contains a distortion element? By this I mean an element $f$ such that the word norm $|f^n|$ grows sublinearly, and $f$ is of infinite order. If such an element exists, what is known about distortion functions (how slowly can the word norm grow)? It is easy to construct subgroups of Brin-Thompson 2V that are distorted in the big group, I am only interested in cyclic ones.
It would suffice to embed a group having distortion elements, for example the discrete Heisenberg group or the Baumslag-Solitar group $\mathrm{BS}(1, 2)$. I am not aware of such an embedding and did not find one on a quick search, but this means very little.
I give one definition of the Brin-Thompson 2V for completeness, this is not the most standard definition (and wouldn't be the first time I manage to mangle it), see any reference for other definitions. The Brin-Thompson 2V is the group acting on $\{0,1\}^{\mathbb{Z}}$ by word-rewritings near the origin. So an element of 2V is a homeomorphism $f : \{0,1\}^{\mathbb{Z}} \to \{0,1\}^{\mathbb{Z}}$ such that you have an in-radius $r$ and a function $\{0,1\}^{2r+1} \to \{0,1\}^* \times {\mathbb{Z}}$, such that if $|u| = r$, $|v| = r+1$ and $f(uv) = (w, n)$, then $f(xu.vy) = \sigma^n(x.wy)$ for any tails $x, y$. Here $\sigma$ is the shift map, $.$ indicates the origin, and juxtaposition is concatenation.
