Roller's problem on median groups

Question

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?

Answer 1

Below are a few examples of groups admitting a median graph as a Cayley graph. (About the terminology "median groups", notice the possible confusion with groups acting geometrically on median spaces.)

• 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.
• C(4)-T(4) presentations. For any C(4)-T(4) presentation whose relations have length four, the corresponding two-complex is a nonpositively curved square complex.
• 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.
• Right-angled mock Artin groups. In the same way that a right-angled Artin group may be associated to a right-angled Coxeter group, Scott introduced right-angled mock Artin groups from right-angled mock reflection groups.
• BMW-groups. In addition to Mark's answer, I would like to mention Caprace's survey Finite and infinite quotients of discrete and indiscrete groups (Section 4, Lattices in products of trees after M. Burger, S. Mozes and D. Wise).

Answer 2

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.