# Roller's problem on median groups

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

A group $G$ is called median if 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?

• Right-angled Artin groups. Given a (simplicial) graph $\Gamma$, define $$A(\Gamma)= \langle \text{vertices of \Gamma} \mid [u,v]=1, \ (u,v) \in E(\Gamma) \rangle.$$ Then the Cayley graph of $A(\Gamma)$ with respect to the generating set given by the previous presentation is median graph, or if you prefer, it is naturally the one-skeleton of a CAT(0) cube complex.
• Right-angled Coxeter groups. The same comment holds for the group $$\langle \text{vertices of \Gamma} \mid u^2=1 \ (\text{u vertex}), \ [u,v]=1 \ (\text{u,v adjacent}) \rangle$$
• LOG groups. Given an oriented graph $\Gamma$ whose edges are labelled by its vertices (such that any vertex labels at most one edge), one can define the group $$G(\Gamma)= \langle \text{vertices of \Gamma} \mid cac^{-1}=b, \ (a \mid c \mid b) \in E(\Gamma) \rangle$$ where $(a \mid c \mid b) \in E(\Gamma)$ means that there exists an oriented edge from $a$ to $b$ which is labelled by $c$. In his paper On the realization of Wirtinger presentations as knot groups, Rosebrock determines precisely when the square complex associated to the above presentation is nonpositively curved.
• Right-angled mock reflection groups. In his paper Right-angled mock reflection and mock Artin groups, Scott classifies explicitely all the groups acting vertex-simply-transitively on CAT(0) cube complexes with $\mathbb{Z}_2$ edge-stabilisers.
There are compact nonpositively-curved square complexes $X$ so that $X$ has a single vertex and the universal cover of $X$ is a CAT(0) square complex isomorphic to the product of two trees, but $\pi_1X$ is not of the form you've mentioned, and doesn't even have a finite-index subgroup admitting such a presentation. For example, there is this paper of Wise.