Gromov's theorem on groups of polynomial growth

One direction is classical algebra with a geometric flavor (Bass–Guivarc'h): if $N \trianglelefteq G$ is nilpotent of finite index with lower central series of ranks $r_1,\dots,r_c$, a counting argument on normal forms of elements shows $b_S(n) \asymp n^d$ with $d = \sum_i i\cdot r_i$, a polynomial bound — so virtually nilpotent groups do have polynomial growth. The converse, due to Gromov, is genuinely deep, and only its strategy is sketched here: rescale the Cayley graph by $1/n$ and take a limit (in the Gromov–Hausdorff sense, along a subsequence and an ultrafilter) as $n \to \infty$; polynomial growth is exactly the condition that keeps these rescaled balls from collapsing to a point or blowing up without bound, so a limiting metric space — the asymptotic cone of $G$ — exists and is a finite-dimensional, locally compact, geodesic space on which $G$ still acts transitively at the level of the limit. Montgomery and Zippin's structure theory of locally compact groups then forces the isometry group of this limit space to contain a Lie group, and a further argument bounds the degree of polynomial growth in terms of the dimension of that Lie group, ultimately pinning down a nilpotent subgroup of finite index inside $G$ itself. A later proof by Kleiner (2010) reroutes this same rescale-and-take-a-limit strategy through the finite-dimensionality of spaces of polynomial-growth harmonic functions, avoiding the Montgomery–Zippin machinery entirely, but the guiding idea — pass to a limit, extract a Lie group, and read off nilpotency — remains the same.