Higher connectedness of asymptotic cones. (Q1398215)

If \((X, d)\) is a metric space and \(s= \{s_1,s_2,\dots\}\) is a sequence of positive numbers approaching \(\infty\), then the scaled spaces \((X, d/s_n)\) need not converge in any natural way to a limit space. However, if one fixes a base point \(1\in X\) and a nonprincipal ultrafilter \(\omega\) on \(\mathbb{N}\), then one can, by set-theoretic trickery, force \((X, d/s_n)\) to converge to a limit space \(\text{Cone}_\omega(X,1,s)\), which we call the asymptotic cone of \(X\). A quasi-isometry \(\Phi: (X, d)\to (Y, d')\) induces a bi-Lipschitz homeomorphism \(\text{Cone}(X,1,s)\to \text{Cone}_\omega(Y,1,s)\) (Proposition 3.5). Hence the algebraic topology of \(\text{Cone}_\omega(X,1,s)\) will supply quasi-isometric invariants of \(X\). The author gives coarse geometric conditions on \((X, d)\) necessary and sufficient for \((X, d)\) to have \(N\)-connected asymptotic cones. These conditions are expressed as coarse filling functions for (coarse) \(N\)-spheres in \(X\). The corollaries yield isoperimetric and isodiametric functions for spaces with \(N\)-connected asymptotic cones. The author proves that asymptotic cones of a virtually polycyclic group are contractible if and only if the group is virtually nilpotent.