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Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.

If there exists a point $p \in O$ such that $$ \lim_{r \to \infty}\frac{\text{Vol} B(p,r)}{r^n} >0, $$ can we prove that $\Gamma$ must be finite?

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    $\begingroup$ Every discrete subgroup $\Gamma$ is virtually isomorphic to $\mathbf{Z}^k$ for some $k$ (which is also the dimension of the unique minimal $\Gamma$-invariant affine subspace $V_\Gamma$), and the given ratio grows as $r^{-k}$, and in particular has positive limsup iff $k=0$, i.e., if $\Gamma$ is finite. I'm just not sure what is the quickest proof. It is useful to have in mind that $\Gamma$ acts cocompactly, and virtually by translations on $V_\Gamma$, but does not always virtually act by translation on all $\mathbf{R}^n$. $\endgroup$
    – YCor
    Feb 14, 2020 at 15:51
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    $\begingroup$ (Unlike what I said in my previous comment, $V_\Gamma$ is not always unique, e.g., when $\Gamma=1$ and $n>0$. Still it is true that $\Gamma$ acts cocompactly on every minimal $\Gamma$-invariant affine subspace.) $\endgroup$
    – YCor
    Feb 15, 2020 at 11:29

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