
Higher connectedness of asymptotic cones
Tim R. Riley Topology, 42, pages 1289–1352, 2003 This article formed a large proportion of my doctoral thesis. We give coarse geometric conditions for a metric space X to have Nconnected asymptotic cones. These conditions are expressed in terms of certain filling functions which are defined recursively on dimension and concern maps to X from the 0skeleta of combinatorial spheres and discs. We prove that the asymptotic cones of X are Nconnected if and only if the kdimensional filling functions are bounded for k=1, 2,..., N+1. We apply this criterion in the case where X is a finitely generated group with a word metric. This leads to information about groups with simply connected cones  in particular they have linear filling length. We prove that if all the asymptotic cones of the group are Nconnected then it is of type F_{N}_{+1} and we provide bounds on Nth order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group are all contractible if and only if the group is virtually nilpotent.
