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.

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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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