At the end of his dissertation *Poc Sets, Median Algebras and Group Actions*, Martin Roller asks

A group $G$ is called

medianif it acts freely and transitively on a median algebra. This is equivalent to saying that the Cayley graph with respect to a certain set of generators is a cubing. [...] Characterize all presentations of median groups.

Do you know any work on the subset? The only examples I am able to imagine are of the form $$\langle x_1, \ldots, x_n \mid [x_i,x_j]=1 \ \text{and} \ x_k^2=1 \ \text{for some} \ 1 \leq i,j,k \leq n \rangle.$$ Are there other simple examples?