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 $w=uv{u}^{-1}{v}^{-1}$ 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 $0\le a\le n$, $0\le b\le n$, and $0\le c\le n^2$ 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 $$\{d{x}^2+d{y}^2+(dz-xdy{)}^2{\}}^{1/2},$$ 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.

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