This dissertation is concerned with the notion of approximation property for discrete quantum groups and in particular weak amenability. Our goal is to apply techniques from geometric group theory to the study of quantum groups. We first give a definition of weak amenability in the setting of discrete quantum groups and we develop some aspects of the general theory, inspired by the classical case. We particularly focus on the notion of Cowling-Haagerup constant. We also define a notion of relative amenability in this context which allows us to prove an additional stability result. Similar results are worked out for the Haagerup property. Eventually, we adress the question of free products of weakly amenable discrete quantum groups. Using the work of E. Ricard and X. Qu on Kintchine inequalities for free products, we prove that if two discrete quantum groups have Cowling-Haagerup constant equal to 1, their free product again has Cowling-Haagerup equal to 1. Secondly, we give examples of weakly amenable discrete quantum groups. To do this, we use the recent work of M. Brannan on the Haagerup property for free quantum groups together with ideas from various works on Haagerup inequalities. More precisely, we give a polynomial bound for the norm of projections on coefficients of an irreducible representation of a free orthogonal quantum groups which allows us to "cut off" M. Brannan's functions and compute the Cowling-Haagerup constant. Finally, we apply techniques of monoidal equivalence to extend these results to other classes of discrete quantum groups, some of which are not unimodular.
