In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from [Formula: see text] into [Formula: see text] ([Formula: see text]) and new insights on the coarse embeddability problem from [Formula: see text] into [Formula: see text], [Formula: see text]. Relevant to geometric group theory purposes, the exact [Formula: see text]-compression of [Formula: see text] is computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.
de la Harpe, P.Topics in geometric group theory (University of Chicago Press, 2000), vi+310 pp., 0 226 31721 8 (paperback), £13.00, 0 226 31719 6 (hardback), £28.50
