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. erpret the criteria in the case where X is a finitely generated group Γ 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 Γ are N-connected then Γ 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 Γ are all contractible if and only if Γ is virtually nilpotent.