** Cubical-like geometry of graph products **

**1. Graph products **

A graph product of groups (do not confuse with graph of groups) is given by a graph, a group for each vertex. The product is the free product of vertex groups modded out so that adjacent vertex groups commute.

**Examples**.

- For a complete graph one gets a direct product.
- If all vertex groups are , one gets right angled Artin groups.
- If all vertex groups are , one gets right angled Coxeter groups.

**2. Quasimedian graphs **

Consider the following generating set: all nonidentity elements of vertex groups. The corresponding Cayley graph is *quasimedian*, meaning that

- graph is connected.
- for any triple of vertices with and , there is a fourth vertex , adjacent to and , such that .
- for any quadruple of vertices such that , and , there is a fifth vertex , adjacent to and , such that .
- and centered are not induced subgraphs.

Quasimedian generalizes *median*, which means quasimedian and triangle-free.

Median graphs are related to CCC: The 1-skeleton of a cube complex is median. Conversely, by filling in cubes in a median graph, one gets a cube complex (Roller, Chepoi).

In a quasimedian graph, a hyperplane is a class of edges for the transitive closure of taking opposite sides of a square or adjacent sides of a triangle.

Theorem 1In a quasimedian graph, consider a hyperplane . Then

- separates,
- the subgraph generated by is isomorphic to where is a clique contained in and is component of the complement of in .
- The distance is a wall distance (counting separating hyperplanes).
- Filling in the graph produces a polyhedral complex.

The only difference between median and quasimedian is that hyperplanes may separate in more than 2 components.

**3. Why study graph products ? **

Combination results exist: properties pass from vertex groups to the product. For instance,

Proposition 2A graph product of groups is .

Indeed, we have a construction of cubical aggregate of spaces. It behaves well for special classes, like cube complexes and special cube complexes.

Proposition 3Let each vertex group act on some quasimedian graph with at least one point with trivial stabilizer. Then the products acts on some quasimedian graph , in such a way that the stabilizer of every maximal clique is a vertex group.

In fact, these combination results extend to arbitrary groups acting on quasimedian with suitable assumptions. The assumptions are restrictive, but they are satisfied in many cases:

- Graph products.
- Diagram products (à la Guba-Sapir).
- Wreath products.
- Certain graphs of groups where vertex groups are direct products.

**4. Geometry of graph products **

Their geometry is easy: distances can be computed.

Assume that the graph and all vertex groups are finite. Then the quasimedian Cayley graph is a good geometric model.

- The product is hyperbolic iff the graph is square-free (Meier).
- The product is virtually cocompact special (Kim).
- The product embeds quasi-isometrically into a product of trees (Dranishnikov, Schröder).

Theorem 4One can decide wether a graph product is relatively hyperbolic or not in the following sense. There is a collection of subgraphs of . The product is relatively hyperbolic iff . If it is, it is relatively to the collection of sub-graph products associated to elements of .