ON THE ASYMPTOTIC HOMOLOGICAL DIMENSION OF HYPERBOLIC GROUPS

We introduce the notion of asymptotic homological dimension asdim(h) of a metric space (invariant under quasiisometry), and show that dim partial derivative(infinity) Gamma+1 less than or equal to asdim(h) Gamma less than or equal to asdim(+) Gamma for a (word-)hyperbolic group Gamma (asdim(+) is the large-scale dimension defined by M. Gromov). We show also that asdim(h) Gamma less than or equal to 2 for a certain class of hyperbolic groups (introduced by M. Gromov) that we call strongly isoperimetric groups.