Example: Gromov-Hausdorff distances and the Heisenberg group, PART 2

As the title shows, this post continues the previous one

Gromov-Hausdorff distances and the Heisenberg group, PART 1

The Heisenberg group G is seen from the point of view of the generating set D. Quantitatively, the group norm “measures how” D generates G. The group norm has the following properties:

  • \| X \| = 0 if and only if X = E = (0,1), the neutral element of G. In general \| X\| \geq 0 for any X \in G.
  • \| X \cdot Y \| \leq \|X\| + \|Y\|, for any X,Y \in G (that is a consequence of the fact that if we want to go from E to X \cdot Y by using horizontal increments, then we may go first from E to X, then from X to X \cdot Y, by using horizontal strings).
  • \| X^{-1} \| = \| X \| for any X \in G (consequence of X \in D implies X^{-1} \in D).

From (group) norms we obtain distances: by definition, the distance between X and Y is

d(X,Y) = \| X^{-1} \cdot Y \|

This is the sub-riemannian distance mentioned at the end of the previous post.

The definition of this distance does not say much about the properties of it. We may use a reasoning similar with the one in (finite dimensional) normed vector spaces in order to prove that any two group norms are equivalent. In our case, the result is the following:

there are strictly positive constants a, c, C such that for any
X \in G (which has the form X = (x, e^{2\pi i z})) with \| X \| \leq a we have

c ( x_{1}^{2} + x_{2}^{2} + \mid z \mid) \leq \|X\|^{2} \leq C ( x_{1}^{2} + x_{2}^{2} + \mid z \mid).

We may take a = 1/3, for example.

For “big” norms, we have another estimate, coming from the fact that the S^{1} part of the semidirect product is compact, thus bounded:

there is a strictly positive constant A such that for any X \in G (which has the form X = (x, e^{2\pi i z})) we have

\| x\| \leq \|X \| \leq \|x\| + A

Let us look now at the ball B(R) = \left\{ X \in G \mbox{ : } \|X\| \leq R \right\} endowed with the rescaled distance

d_{R} (X,Y) = \frac{1}{R} d(X,Y)

Denote by Profile(R) = [B(R), d_{R}] the isometry class (the class of metric spaces isometric with … ) of (B(R), d_{R}). This is called a “metric profile”, see Introduction to metric spaces with dilations, section 2.3, for example.

The function which associates to R > 0 the Profile(R) can be seen as a curve in the Gromov space of (isometry classes of) compact metric spaces, endowed with the Gromov-Hausdorff distance.

This curve parameterized with R roams in this huge abstract space.
I want to see what happens when R goes to zero or infinity. The interpretation is the following: when R is small (or large, respectively), how the small (or large) balls look like?

Based on the previous estimates, we can answer this question.

When R goes to infinity, the profile Profile(R) becomes the one of the unit ball in \mathbb{R}^{2} with the euclidean norm. Indeed, this is easy, because of the second estimate, which implies that for any X = (R x, e^{2 \pi i z}) and Y = (R y, e^{2 \pi i u}) which belong to B(R), (thus \|x\|, \|y\| \leq 1) we have:

d_{euclidean}(x, y) \leq d_{R}(X,Y) \leq d_{euclidean}(x, y) + \frac{A}{R}.

Therefore, as R goes to infinity, we get the isometry result.

On the other side, if R is small enough (for example smaller or equal to 1/3, then Profile(R) becomes stationary!

Indeed, let me introduce a second Heisenberg group, baptized H(1) = \mathbb{R}^{2} \times R, with the group operation

(x, z) \cdot (y, u) = (x+ y, z + u + \frac{1}{2}\omega(x,y))

Remark that the function (x, e^{2 \pi i z}) \mapsto (x,z) is a group morphism (in fact a local group isomorphism), for z small enough! That means locally the groups G and H(1) are isomorphic. If you don’t know what a local group is then see the post Notes on local groups by Terence Tao.

By exactly the same procedure, we may put a group norm on H(1).

OK, so small balls in G are isometric with small balls in H(1). What about the rescaling with \frac{1}{R}? Well, it turns out that the group H(1) is selfsimilar, moreover, is a conical group (see for example section 6 from the paper Braided spaces with dilations… and check also the references, for the notion of conical group). Conical means that the group has a one parameter family of self-similarities: for any R > 0 the function

\delta_{R} (x,z) = (R x, R^{2} z)

is an auto-morphism of H(1) and moreover:

\| \delta_{R} (x,z) \| = R \| (x,z)\| for any (x,z) \in H(1).

As a consequence, all balls in H(1) look alike (i.e. the metric profile of the group H(1) is stationary, a hallmark of null curvature…). More precisely, for any R > 0 and any X,Y \in H(1), if we denote by d the distance in H(1) induced by the group norm, we have:

d_{R}( \delta_{R} X, \delta_{R} Y) = d(X,Y).

Conclusion for this part: Small balls in G look like balls in the Heisenberg group H(1). Asymptotically, as R goes to infinity, balls of radius R in the group G look more and more alike balls in the euclidean space \mathbb{R}^{2} (notice that this space is self-similar as well, all balls are isometric, with distances properly rescaled).


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