 # 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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