We prove some rather precise renorming theorems for Banach spaces with Szlenk index ω0. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

Let M be a triangulable compact manifold. We prove that, among closed subgroups of Homeo0(M) (the identity component of the group of homeomorphisms of M), the subgroup consisting of volume preserving elements is maximal. AMS classification. 57S05 (57M60, 37E30).