The first time I heard about the asymptotic cone, I ingenuously thought "Well... the asymptotic cone of $\mathbb Z^2$ minus the origin is $\mathbb R^2$ minus the origin". At that point somebody said me "Are you crazy? The asymptotic cone is always a geodesic space! The asymptotic cone of $\mathbb Z^2$ minus one point is $\mathbb R^2$".
This is seems very strange to my (certainly questionable) intuition. Trying to imagine myself moving further and further from $\mathbb Z^2$ minus the origin, I would say the hole gets smaller and smaller... why does it disappear? I would say that it gets infinitesimal...
Question: Is there any similar notion of asymptotic cone, taking into account, in some sense, the holes?
Why I am interested in this?
First of all because I am human being and, even if I am male, I am quite curious. Second, because I think that there might be some interesting relation with the A-theory of graphs. For instance, the fundamental group of $\mathbb Z^2$ minus one point is equal to $\mathbb Z$ in A-theory. The $A$-theory is really constructed as the discretization of the standard homotopy theory and then, conversely, it is possible that the homotopy theory of the continuous version of a discrete object (as the asymptotic cone in my intuition should be) is related to the A-theory of the object itself. Third because, if the classical asymptotic cone is useful in many cases, maybe a more precise notion as the one I am trying to imagine, can be even more useful... (or maybe not, since such a notion cannot be a quasi-isometric invariant).
Thanks in advance,