Metric approach to limit operators VI

I thought I’d finished this sequence of posts with number five, but then I spent a little time talking with Jerry Kaminker and Rufus Willett and I think that I understood two things: first, how to formulate the limit point construction more cleanly and, second, the “symmetry breaking” role of the ultrafilters which is not clear in what I had written so far.  Read on.

Understanding the limit space construction.  Suppose that $$X$$ is a bounded geometry, uniformly discrete metric space.  It has a “geometry bound”, then, which is a function $$g$$ such that every ball of radius $$R$$ contains at most $$g(R)$$ points.  If we think about the possible balls of a fixed radius $$R$$ in such a space, then, they are determined (as metric spaces) by at most $$g(R) ( g(R)-1)/2$$ distances each of which lies between $$1$$ and $$2R$$; it follows that the space of such balls is itself a compact (metric) space $${\mathfrak M}(g,R)$$.  Let

${\mathfrak M}(g) = \prod_{n=1}^\infty {\mathfrak M}(g,n)$

which is also a compact metric space.  Then for any $$X$$ with the given geometry bound, and any $$x\in X$$, we can map $$X$$ to $${\mathfrak M}(g)$$ by taking the sequence of balls $$B(x;n)$$, and we can define the distance between two such spaces $$X$$ and $$X’$$ to be the distance between their images in $${\mathfrak M}(g)$$.  Of course, this is simply a version of the pointed Gromov-Hausdorff distance.  By construction, the space $${\mathfrak N}(g)$$ of uniformly discrete pointed metric spaces with geometry bound $$g$$ is then a compact metric space in the Gromov-Hausdorff metric.

Now, the punchline: There is an obvious map

$X \to {\mathfrak N}(g)$

which sends each $$x\in X$$ to the pointed metric space $$(X,x)$$.  By the universal property of the Stone-Cech compactification, any such map from $$X$$ to a compact metric space extends uniquely  to a continuous map defined on $$\beta X$$.  In particular we obtain a continuous map $$\partial X \to {\mathfrak N}$$, and this is the map which assigns the limit space $$X_\omega$$ to a free ultrafilter $$\omega$$. The continuity of $$X_\omega$$ as a function of $$\omega$$, which I was laboriously explaining in the previous post, is now obvious.

The tangent bundle As we pointed out earlier, the assignment $$\omega\mapsto X_\omega$$ should be thought of as some kind of “bundle of metric spaces” over $$\partial X$$.  But this is not an arbitrary such bundle.  As seen in the S-W original construction, there is a natural identification of $$X_\omega$$ with a (very tiny, and not open) “neighborhood” of $$\omega$$ in $$\partial X$$. What kind of bundle, in ordinary geometry, has this property that its fibers are naturally identified with neighborhoods of their basepoint?  It’s suggestive to say that what S-W have defined is the “metric tangent bundle” to the Stone-Cech compactification!

Symmetry breaking.  It’s important for the “condensation of singularities” part of the argument to keep track, not only of the fact that balls $$B(x; R)$$ in $$X$$ become (nearly) isometric to $$B(\omega; R)$$ in $$X_\omega$$ as $$x$$ approaches $$\omega$$, but also of the particular (near) isometries that implement this.   For example, if $$X = {\mathbb Z}$$, then the limit space is $$\mathbb Z$$ as well and there are two ways to identify an interval in $$X$$ with the corresponding interval in the limit space – one orientation-preserving and one orientation-reversing isometry.  (I had not fully appreciated the importance of this point in the posts I wrote earlier, but after Rufus illustrated it in his talk with expansive hand gestures, it became impossible to ignore.)  In the discussion above this corresponds to taking $${\mathfrak M}(g,R)$$ to be (a subspace of) a product of intervals (as I defined it), rather  than the quotient of that object by the natural action of the symmetric group.  We can think of this as a Gromov-Hausdorff space of well ordered (countable) metric spaces – the ordering is arbitrary, it is just used to keep track of which point is #1, which is #2 and so on…  But the resulting GH space is still compact, and $$X_\omega$$ is taken to be an object of this kind.  What that means is that there is given a specific choice of identification of the ball $$B(\omega;r)\subseteq X_\omega$$ with $$B(x;r)\subseteq X$$, for an $$\omega$$-dense set of $$x$$ in $$X$$.