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}}.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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