Apart from the abstract types of the crystallographic groups, are there any other abstract groups that admit a proper, co-compact, uniformly bilipschitz action on $\mathbb{R}^3$?



Infinite groups that admit a discrete, co-compact, bilipschitz action on $\mathbb{R}^3$

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> Fix $k\ge 0$. Let $\Gamma$ be a discrete group. Then $\Gamma$ (a) has a proper cocompact, uniformly bilipschitz action on the Euclidean space $\mathbf{R}^k$ if and only if (b) it has an isometric one, if and only if (c) it has a finite index subgroup isomorphic to $\mathbf{Z}^k$.
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Proof: That (c) implies (b) is very standard and (b) trivially implies (a). Suppose (a): consider such an action $(g,x)\mapsto gx$, assume each map $(1/C,C)$-bilipschitz. Fix $x$ in $\mathbf{R}^k$ and a symmetric generating subset $S$ of $\Gamma$. Write $M=\max\_{s\in S}d(x,sx)$ and $C'=CM$.


Then $g\mapsto gx$ is a quasi-isometry. For $g,h\in\Gamma$, let $n=d\_S(g,h)$ be their word distance: consider elements $g\_0,\dots,g\_n$ with $g\_0=g$, $g\_n=h$, $d(g\_i,g\_{i+1})\le 1$. Then


$$d(gx,hx)\le \sum\_{i=0}^{n-1} d(g\_ix,g\_{i+1}x)\le C\sum\_{i=0}^{n-1} d(x,g\_i^{-1}g\_{i+1}x)\le nCM=C'd\_S(g,h),$$


and $d(gx,hx)\ge C^{-1}d(x,g^{-1}h)$ which tends to infinity when $d\_S(g,h)$ tends to infinity.


So $x\mapsto gx$ Lipschitz, uniformly proper, and has cobounded image and is a map between geodesic spaces. Hence it is a quasi-isometry.


We conclude by using that a group that is quasi-isometric to $\mathbf{R}^k$ satisfies (c). (This is a standard consequence of Gromov's theorem on groups with polynomial growth, e.g. using Pansu's theorem, and also admits proofs not appealing to Gromov's theorem.)


