## Notes of Anthony Genevois’ Cambridge lecture 10-01-2017

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 ${{\mathbb Z}}$, one gets right angled Artin groups.
• If all vertex groups are ${{\mathbb Z}/2{\mathbb Z}}$, 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 ${a,b,c}$ of vertices with ${ab=ac=k}$ and ${bc=1}$, there is a fourth vertex ${w}$, adjacent to ${b}$ and ${c}$, such that ${aw=k-1}$.
• for any quadruple ${a,b,c,d}$ of vertices such that ${ab=ad=k}$, ${bc=cd=1}$ and ${ac=k+1}$, there is a fifth vertex ${w}$, adjacent to ${b}$ and ${d}$, such that ${aw=k-1}$.
• ${\Theta}$ and centered ${\Theta}$ are not induced subgraphs.

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

Median graphs are related to CCC: The 1-skeleton of a ${CAT(0)}$ cube complex is median. Conversely, by filling in cubes in a median graph, one gets a ${CAT(0)}$ 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 1 In a quasimedian graph, consider a hyperplane ${J}$. Then

1. ${J}$ separates,
2. the subgraph generated by ${J}$ is isomorphic to ${F\times C}$ where ${C}$ is a clique contained in ${J}$ and ${F}$ is component of the complement of ${J}$ in ${N(J)}$.
3. The distance is a wall distance (counting separating hyperplanes).
4. Filling in the graph produces a ${CAT(0)}$ 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 2 A graph product of ${CAT(0)}$ groups is ${CAT(0)}$.

Indeed, we have a construction of cubical aggregate of ${CAT(0)}$ spaces. It behaves well for special classes, like ${CAT(0)}$ cube complexes and special ${CAT(0)}$ cube complexes.

Proposition 3 Let each vertex group ${G_v}$ act on some quasimedian graph ${X_v}$ with at least one point with trivial stabilizer. Then the products acts on some quasimedian graph ${Y}$, 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:

1. Graph products.
2. Diagram products (à la Guba-Sapir).
3. Wreath products.
4. 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.

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

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