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 \(X\).

 

 

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