Notes on Asymptotic Cones

Let X be a metric space, {xi}i∈N a sequence of points in X (thought of as a sequence of centers), and {di}i∈N a sequence of scaling factors such that di → ∞, and an ultrafilter ω. I’m not going to get into the details of ultrafilters here, but the main property of an ultrafilter is that it lets you define the ultralimit limω ai of any bounded sequence ai. Ultralimits use the axiom of choice to extend the notion of limit to arbitrary bounded sequences; they are linear, and the ultralimit of a sequence is a limit point of the sequence. We use these to construct an asymptotic cone. Let XN b be the set of sequences {yi}i∈N such that d(xi, yi)/di is bounded. This looks huge, but it’s like defining real numbers by Cauchy sequences – a lot of sequences have the same limit. Define dω({xi}, {yi}}) = limω d(xi, yi)/di, and define Conω X = XN b / ∼ {xi} ∼ {yi} iff dω({xi}, {yi}) = 0. This is a metric space; we call it an asymptotic cone for X. Denote the equivalence class of {yi}i∈N by [yi]. Alternatively, this is the Gromov-Hausdorff limit of a sequence of scalings of X with basepoint xi.