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.
- 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 1 In a quasimedian graph, consider a hyperplane . Then
- 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 2 A 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 3 Let 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 4 One 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 .