## Notes of Goulnara Arzhantseva’s second Oxford lecture 21-03-2017

Constructions of non ${C^*}$-exact groups, II

1. Embedding graphs in groups

1.1. Almost quasi-isometric embeddings

Say a sequence of maps ${f_n:X_n\rightarrow Y}$ constitute an almost quasi-isometric embedding if

$\displaystyle \begin{array}{rcl} A\,d_n(x,x')-B_n\leq d(f(x),f(x'))\leq C\,d_n(x,y), \end{array}$

where ${B_n=o(}$girth${(X_n))}$.

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

2. Graphical presentations

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

Example. If ${X=Cay(G,S)}$, then ${G(X)=G}$.

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

2.1. Quotients from coverings

Let ${p:Y\rightarrow X}$ be a label-preserving graph covering. This induces a surjective homomorphism ${G(Y)\rightarrow G(X)}$ (can be traced back to von Dyck in 1882).

If ${X}$ is a Cayley graph, ${X=Cay(G,S)}$ and if ${\pi_1(Y)}$ is not merely normal, but a characteristic subgroup in ${\pi_1(X)}$, then ${Y}$ is a Cayley graph too, ${Y=Cay(H,S)}$ where ${H}$ surjects onto ${G}$. 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 ${X}$ to embed in ${Cay(G(X),S)}$.

Say a presentation is ${C'(\lambda)}$ if every piece (common part between two relators) satifies ${|}$piece${|\leq\lambda|}$relator.

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. ${G=\langle a,b|au_1,bv_1,au_2,bv_2,\ldots\rangle}$, where ${u_n=(a^n b^n)^{10}}$, ${v_n=(a^n b^{2n})^{10}}$, all ${n\geq 1}$. 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 ${X=\bigcup X_n}$ be a ${S}$-labelled graph, with reduced labelling, satisfying a suitable, geometric, small cancellation condition. Let ${H}$ be a non-elementary hyperbolic group. Then ${X_0}$ embeds almost quasi-isometrically into ${Cay(H(X),S)}$.

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 ${X=\bigcup X_n}$ be a ${S}$-labelled graph, with reduced labelling., satisfying graphical ${C'(1/6)}$ condition. Then ${X_0}$ embeds injectively and isometrically into ${Cay(G(X),S)}$.

The proof of injectivity is very similar to classical ${1/6}$-cancellation: Euler formula forbids ${(3,7)}$ 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 ${X=\bigcup X_n}$ be a large girth dg-bounded graph (i.e. diameter ${=O(}$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: ${Cay(Sl(2,F_p),S)}$ where ${S}$ contains ${\begin{pmatrix} 1&2\\0&1 \end{pmatrix}}$ 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.