Equivariant covers for hyperbolic groups

Recall that a cover U is of dimension N if every x 2 X is contained in no more then N C 1 members of U . The asymptotic dimension of a finitely generated group is its asymptotic dimension as a metric space with respect to any word metric. An important result of Yu [19] asserts that the Novikov conjecture holds for groups of finite asymptotic dimension. This can be viewed as an injectivity result for the assembly map in L–theory (after inverting 2). Further injectivity results for assembly maps for groups with finite asymptotic dimension can be found in Bartels [1], Carlsson and Goldfarb [6] and Bartels and Rosenthal [4]. On the other hand no surjectivity statement of assembly maps is known for all groups of finite asymptotic dimension and this is very much related to the absence of any equivariance condition for the cover U as above.