Consider again the three-dimensional Heisenberg group $H$ with its general element written $$\left(\begin{array}{ccc}1& x& z\\ 0& 1& y\\ 0& 0& 1\end{array}\right).$$ Although it is thereby identified with ${\mathbb{R}}^3$, we find that if we want left-multiplications in the group to be isometries we must warp the usual metric of ${\mathbb{R}}^3$ somewhat. In fact the length of a path must be defined as the integral of $$\{d{x}^2+d{y}^2+(dz-xdy{)}^2{\}}^{1/2},$$ which means that each plane perpendicular to the $x$-axis has been sheared in the $y$-direction by an amount proportional to $x$. As we re-scale the word-metric on $H_{\mathbb{Z}}$ we must re-scale the metric on ${\mathbb{R}}^3$ too, multiplying $x$ and $y$ by $\lambda =1/r_i$, and $z$ by ${\lambda }^2$. In the limit, this means that the only paths of finite length are those whose direction at each point lies in the plane given by $dz-xdy=0$. Let us call them the allowable paths. They give us a new metric on ${\mathbb{R}}^3$ in which the distance from one point to another is the length of the shortest allowable path joining them. This metric defines the usual topology on ${\mathbb{R}}^3$, but, unlike the taxi-cab metric, it is far from equivalent --- even locally --- to the usual metric. Nevertheless, it is a metric which those of us who have tried to park a car know all too well. To move a car just a little bit sideways we must take it along a disproportionately long path. This is because the position of a car sitting on an expanse of tarmac is described by three coordinates $(\xi ,\eta ,\theta )$, where $(\xi ,\eta )$ is the position of the mid-point of the front wheels, and $\theta$ is the angle in which the axis of symmetry of the car is pointing. When we want to move the car we can move $\xi$ and $\eta$ any way we like, within reason, but the change in the third coordinate is constrained by the differential relation $d\theta =\cos \theta d\eta -\sin \theta d\xi ,$ assuming the car is of unit length. (A simple change of coordinates puts this in the form $dz=ydx$ which we found for the group $H$.) What is remarkable about this new metric is that on one hand it defines the usual topology of ${\mathbb{R}}^3$, but on the other hand it defines a metric space of Hausdorff dimension four. Hausdorff dimension is a concept defined only for metric spaces. To say it is four means, essentially, that the number of balls of radius $r/n$ needed to cover a ball of radius $r$ grows like $n^4$ as $n\to \mathrm{\infty }$. In the new metric on ${\mathbb{R}}^3$ this is the case because a small $\epsilon$-ball for the new metric looks (in the usual coordinates) like a very flat ellipsoid with axes $(\epsilon ,\epsilon ,{\epsilon }^2)$. This explains the quartic growth rate of the group $H_{\mathbb{Z}}$. More generally, for any metric space the topological dimension is bounded above by the Hausdorff dimension, and so the finite-dimensionality of the limit space $Y$ follows from the polynomial growth of the group.

Return to main article.