Cayley graphs containing expanders, after Gromov

We give the sketch of a combinatorial proof of the construction by Gromov of a group whose Cayley graph contains a family of expanders. Combining the methods in [Oll1] and [Oll2], it is possible to give a proof of the construction invented by M. Gromov in [G] that there is an infinite group whose Cayley graph contains (in some quasi-isometric sense) a family of expanders. We only give the main steps of the proof, as our goal is to illustrate our techniques and not to re-prove known theorems. This text is a natural sequel to [Oll2], and also heavily relies on sections 5.1.1, 6.2 to 6.6 and Appendix A of [Oll1]. The main lines of the argument of [G] are also explained in [Gh]. 1 Quotients of hyperbolic groups by labelled graphs We give here a statement of a theorem generalizing the one stated in [Oll2], together with a sketch of proof. We use the terminology of [Oll2]: Γ is a graph labelled with the generators a±1 1 , . . . , a ±1 m of some group G0, and we want to study the quotient of G0 by the words read on cycles of the graph. In this more complex situation, an ε-piece with respect to G0 is a couple of words (w1, w2) embedded in Γ (not necessarily distinct) together with a couple of words (δ1, δ2), such that |δ1| + |δ2| 6 ε(|w1| + |w2|) and such that w1 = δ1w2δ2 in G0. Again we have to eliminate trivial cases: for example, if the word uwv is embedded in Γ, then ((uw,wv), (u, v−1)) is a trivial piece. Generally, a trivial piece is a piece ((w1, w2), (δ1, δ2)) such that there is a path in Γ joining the beginning of w1 to the beginning of w2, labelling a word equal to δ1 in G0. The length of a piece is defined as max(|w1| , |w2|). We will say that a group is aspherical if it admits a presentation with no spherical van Kampen diagram (with the convention of [Oll1] for van Kampen