Higher connectedness of asymptotic cones

We give coarse geometric conditions for a metric space X to have N-connected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling N-spheres in an appropriate coarse sense. We interpret the criteria in the case where X is a finitely generated group Gamma with a word metric. This leads to upper bounds on filling functions for groups with simply connected cones-in particular they have linearly bounded filling length functions. We prove that if all the asymptotic cones of Gamma are N-connected then Gamma is of type FN+1 and we provide Nth order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group Gamma are all contractible if and only if Gamma is virtually nilpotent.