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).