** Constructions of non -exact groups, II **

**1. Embedding graphs in groups **

** 1.1. Almost quasi-isometric embeddings **

Say a sequence of maps constitute an almost quasi-isometric embedding if

where girth.

We shall see that such embeddings from graphs to groups are rather easy to construct.

**2. Graphical presentations **

Definition 1 (Rips-Segev 1987)Let be a finite set of labels. Let be an -labelled graph. Let be the group with generating system and as relators all words read along non-trivial simple closed paths in .

**Example**. If , then .

Our goal is rather to produce group from graph. If is a bouquet of circles, one recovers usual presentations.

** 2.1. Quotients from coverings **

Let be a label-preserving graph covering. This induces a surjective homomorphism (can be traced back to von Dyck in 1882).

If is a Cayley graph, and if is not merely normal, but a characteristic subgroup in , then is a Cayley graph too, where surjects onto . Indeed, a graph is a Cayley graph iff it is a Galois covering of a bouquet of circles.

** 2.2. Small cancellation conditions **

These are sufficient condition for the graph to embed in .

Say a presentation is if every piece (common part between two relators) satifies piecerelator.

The notion can be adapted to graphical presentations: pieces are labelled path common to two labelled cycles. Gromov even extends this to producing quotients of arbitrary hyperbolic groups. Delzant and I have developped this aspect.

Dominik Gruber observed that one could take pieces and cycles up to graph automorphism.

*Example*. , where , , all . This is an infinite usual presentation, but a finite graphical presentation. It is not residually finite.

** 2.3. Small cancellation theorem **

Theorem 2 (Gromov 2003, Arzhantseva-Delzant 2008)Let be a -labelled graph, with reduced labelling, satisfying a suitable, geometric, small cancellation condition. Let be a non-elementary hyperbolic group. Then embeds almost quasi-isometrically into .

The proof of this geometric version relied of the local to global Cartan-Hadamard theorem for hyperbolic groups.

Theorem 3 (Gromov 2003, Ollivier 2005, Gruber 2012)Let be a -labelled graph, with reduced labelling., satisfying graphical condition. Then embeds injectively and isometrically into .

The proof of injectivity is very similar to classical -cancellation: Euler formula forbids cellulations of planar disks, hence arcs cannot be mapped to loops.

Proof of isometry uses Strebel’s classification of bigons.

** 2.4. Small cancellation labellings **

How does one choose labellings ?

Theorem 4Let be a large girth dg-bounded graph (i.e. diameter girth). Then with overwhelming propability, a randomly chosen labelling has geometric cancellation (Gromov 2003). With positive probability, it satisfies graphical small cancellation (Osajda 2014).

Fortunately, there exist explicit large girth dg-bounded graphs, see Lubotzky-Phillips-Sarnak 1988. Although the theory is hard, there is an easily describable example: where contains and its transpose.

** 2.5. Special Gromov monsters **

This completes the construction of Gromov monsters (geometric case) and *special Gromov monsters* (graphical case).

Special Gromov monsters enjoy the following property:

Theorem 5 (Gruber-Sisto 2014)Graphical small cancellation groups are acylindrically hyperbolic.

Next time, we shall see how to construct true monsters, which have additionnally Haagerup’s property.