The Novikov Conjecture for Groups with Finite Asymptotic Dimension

Recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology (page 28, [14]). More precisely, the asymptotic dimension for a metric space is the smallest integer n such that for any r > 0, there exists a uniformly bounded cover C = {Ui}iEI of the metric space for which the r-multiplicity of C is at most n + 1; i.e., no ball of radius r in the metric space intersects more than n + 1 members of C [14]. The class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word-length metric, then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word-length metrics (cf. Section 6). This, together with a result of Gromov in [14], implies that finitely generated subgroups of Gromov's hyperbolic groups have finite asymptotic dimension. Currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known. It should also be noted that two different definitions of asymptotic dimension