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co.combinatorics – When does a subgroup of $operatorname{GL}(n, mathbb Z)$ have a bounded basic province on $mathbb R^n$? Answer

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co.combinatorics – When does a subgroup of $operatorname{GL}(n, mathbb Z)$ have a bounded basic province on $mathbb R^n$?

$DeclareMathOperatorGL{GL}$Let $G$ breathe a finitely generated subgroup of $GL(n,mathbb Z)$. Then $G$ acts on $mathbb R^n = mathbb Z^n otimes_{mathbb Z} mathbb R$ by means of $GL(n,mathbb Z)$.

Suppose that there’s a rational affine subspace $V subset mathbb R^n$ (by this, I denote that there’s a sub-lattice $L subset mathbb Z^n$ and $a in mathbb Q^n$ such that $V = a + (L otimes_{mathbb Z} mathbb R)$), and $V$ is invariant beneath the motion of $G$ (i.e. for any $vin V, gin G$, we’ve got $gcdot v in V$). Moreover, there exists $v in L$ such that
$$G cdot v = L.$$

Question: is there a bounded subset $P subset V$ such that $$bigcup_{g in G} gcdot P = V quad ? $$

Any suggestion on pertinent questions/references may be very welcome! Particularly, I do not know which bailiwick research such issues ….

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