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
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 4 Let 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.