The simplest non-Abelian example

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It is worth considering the simplest non-abelian example more closely. The integer Heisenberg group $H_{\mathbb Z}$ is the simplest non-trivial example of a nilpotent group. It is generated by two elements $u,v$ with the relations that the commutator $w = uvu^{-1}v^{-1}$ commutes with $u$ and $v$. Explicitly, it is the group of $3\times 3$ upper-triangular matrices with integer entries and diagonal entries 1: take $u$ and $v$ to be the matrices

  \[ u = \left( \begin{array}{ccc} 1 &  1 &  0 \\ 0 &  1 &  0 \\ 0 &  0 &  1 \end{array} \right),\   v = \left( \begin{array}{ccc} 1 &  0 &  0 \\ 0 &  1 &  1 \\ 0 &  0 &  1 \end{array} \right), \]    

so that

  \[ w = \left( \begin{array}{ccc} 1 &  0 &  1 \\ 0 &  1 &  0 \\ 0 &  0 &  1 \end{array} \right). \]    

Any element of $H_{\mathbb Z}$ can be written uniquely in the form $u^ k v^ l w^ m$ for some integers $k,l,m$.

The group $H_{\mathbb Z}$ sits inside the more usual 3-dimensional Heisenberg group $H$ consisting of the strictly upper-triangular matrices with real entries

  \[ \left( \begin{array}{ccc} 1 &  x &  z \\ 0 &  1 &  y \\ 0 &  0 &  1 \end{array} \right). \]    

By analogy with the case of $\mathbb Z^2$ in the plane, you might expect the group $H_{\mathbb Z}$ to converge to $H$ under re-scaling. As a manifold, the group $H$ is just $\mathbb R^3$, so you would predict that $H_{\mathbb Z}$ has cubic polynomial growth just like the abelian group $\mathbb {Z}^3$. But actually it has quartic growth. This is easy to see: because $vu = uvw$ we have $v^ m u^ n = u^ n v^ m w^{mn}$ for any $m,n$, and so any of the $n^4$ elements $u^ a v^ b w^ c$ with $0 \leq a \leq n$, $0 \leq b \leq n$, and $0 \leq c \leq n^2$ can be obtained as the product of a string of at most $n$ copies of $u$ and at most $n$ copies of $v$ arranged in a suitable order. Why do we get quartic growth from such a 3-dimensional group? The answer takes us to a fascinating piece of geometry.

Although $H$ is 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

  \[ \{ dx^2 \  + \  dy^2 \  + \  (dz - xdy)^2 \} ^{1/2}, \]    

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 \infty $. In the new metric on $\mathbb R^3$ this is the case because a small $\varepsilon $-ball for the new metric looks (in the usual coordinates) like a very flat ellipsoid with axes $(\varepsilon , \varepsilon , \varepsilon ^2)$. 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.

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