A metric space $(M,d)$ is *doubling* if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric generating sets $S$, is $\mathrm{Cay}(G, S)$ doubling under the path metric? Groups like this have polynomial growth, so they are virtually nilpotent by Gromov's theorem.


So which virtually nilpotent groups are doubling, and for which generating sets? All, I suppose, but I got cold feet trying to do it, it seemed quite difficult straight from the definitions and I don't really know the Lie group stuff well enough.



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> If $S$ is a finite symmetric generating set for a group $G$, is $\mathrm{Cay}(G, S)$ doubling precisely when $G$ is virtually nilpotent?
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I'll note that in general (undirected) graphs, doubling implies polynomial growth, but not the other way around, consider for example the comb graph with vertices $\mathbb{Z} \times \mathbb{N}$ and edges $\{\{(m,n), (m,n+1)\}, \{(m,0), (m+1,0)\} \;|\; m \in \mathbb{Z}, n \in \mathbb{N}\}$. But could be true for vertex-transitive graphs.



Which groups are doubling?
I think this follows from a standard ball-packing argument.


Suppose that $G$ with the metric $\rho$ induced from the Cayley graph has growth $V(R)=|B\_R(1)| \sim R^d$, i.e. $\exists\ 0<c< C$ such that $cR^d\leq V(R)\leq CR^d$, where $B\_R(1)$ is the open ball of radius $R$ about the identity (indeed, this argument works for any metric-measure space with polynomial growth of balls about every point in this sense). This holds for finitely generated nilpotent groups
by a [result of Bass](https://mathscinet.ams.org/mathscinet-getitem?mr=379672).


Then take a maximal [$R/2$-packing](https://en.wikipedia.org/wiki/Delone_set) $N\_R$ of $B\_R(1)$, i.e. $N\_R\subset B\_R(1)$ where $\rho(g\_1,g\_2)\geq R/2$ for $g\_1,g\_2\in N\_R, g\_1\neq g\_2$. Then $B\_{R/4}(g\_1)\cap B\_{R/4}(g\_2) =\emptyset$.


By maximality, $B\_R(1)\subseteq \cup\_{g\in N\_R} B\_{R/2}(g)$: if not, then we could find another point $h\in B\_R(1)$ whose distance $\rho(h,N\_R)$ is at least $R/2$, so $N\_R\cup \{h\}$ is an $R/2$-packing, a contradiction to maximality of $N\_R$. Thus $N\_R$ is an $R/2$-net of $B\_R(1)$.


Moreover, the union of the balls of radius $R/4$ about points of $N\_R$ lies in $B\_{5R/4}(1)$. Hence $|N\_R|V(R/4) \leq V(5 R/4)$. So we have $|N\_R| \leq \frac{V(5R/4)}{V(R/4)} \leq \frac{C (5R/4)^d}{c(R/4)^d} =C/c5^d $. Hence the space is doubling.


