Algèbres de Royden et homéomorphismes à p-dilatation bornée entre espaces métriques mesurés

The goal of this thesis is to introduce the notions of Royden algebra and mapping with bounded p-dilation between metric measure spaces. In particular, we give sufficient conditions for a metric measure space to be characterized, up to bilipschitz equivalence, by its Royden algebra. We also study sufficient conditions for mapping with bounded p-dilation between metric measure spaces to be a quasi-isometry or to be a lipschitiz map. Our results are obtained in the framework of the theory of axiomatic Sobolev spaces on metric measure spaces and are thus of a very general nature. However, we show that the application of these results to the particular case of nilpotent Lie groups with a Hörmander system gives new concrete information on the geometry of these groups.