As the title shows, this post continues the previous one
The Heisenberg group is seen from the point of view of the generating set . Quantitatively, the group norm “measures how” generates . The group norm has the following properties:
- if and only if , the neutral element of . In general for any .
- , for any (that is a consequence of the fact that if we want to go from to by using horizontal increments, then we may go first from to , then from to , by using horizontal strings).
- for any (consequence of implies ).
From (group) norms we obtain distances: by definition, the distance between and is
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 such that for any
(which has the form ) with we have
We may take , for example.
For “big” norms, we have another estimate, coming from the fact that the part of the semidirect product is compact, thus bounded:
there is a strictly positive constant such that for any (which has the form ) we have
Let us look now at the ball endowed with the rescaled distance
Denote by the isometry class (the class of metric spaces isometric with … ) of . This is called a “metric profile”, see Introduction to metric spaces with dilations, section 2.3, for example.
The function which associates to the 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 roams in this huge abstract space.
I want to see what happens when goes to zero or infinity. The interpretation is the following: when is small (or large, respectively), how the small (or large) balls look like?
Based on the previous estimates, we can answer this question.
When goes to infinity, the profile becomes the one of the unit ball in with the euclidean norm. Indeed, this is easy, because of the second estimate, which implies that for any and which belong to , (thus ) we have:
Therefore, as goes to infinity, we get the isometry result.
On the other side, if is small enough (for example smaller or equal to , then becomes stationary!
Indeed, let me introduce a second Heisenberg group, baptized , with the group operation
Remark that the function is a group morphism (in fact a local group isomorphism), for small enough! That means locally the groups and 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 .
OK, so small balls in are isometric with small balls in . What about the rescaling with ? Well, it turns out that the group 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 the function
is an auto-morphism of and moreover:
for any .
As a consequence, all balls in look alike (i.e. the metric profile of the group is stationary, a hallmark of null curvature…). More precisely, for any and any , if we denote by the distance in induced by the group norm, we have:
Conclusion for this part: Small balls in look like balls in the Heisenberg group . Asymptotically, as goes to infinity, balls of radius in the group look more and more alike balls in the euclidean space (notice that this space is self-similar as well, all balls are isometric, with distances properly rescaled).