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

$DeclareMathOperatorGL{GL}$Let $G subset M_{ntimes n~}(mathbb Z)$ breathe a **finitely generated** subgroup of $GL(n,mathbb Q)$ (i.e. $gin G$ is an invertible matrix with entries in $mathbb Z$). Then $G$ acts on $mathbb R^n = mathbb Z^n otimes_{mathbb Z} mathbb R$ via $GL(n,mathbb Q)$.

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 $a in mathbb Z^n$ such that $V = a + (L otimes_{mathbb Z} mathbb R)$), and $V$ is invariant underneath the motion of $G$ (i.e. for any $vin V, gin G$, now we have $gcdot v in V$). Moreover, there exists $v in L$ (in truth, we are able to take $v=a$) 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 ….

**Edit:**

**Example.** Consider $(0,1)+L:=(mathbb Z,1) subset mathbb Z^2$, and $$G={commence{pmatrix} 1&okay&1end{pmatrix}mid kinmathbb Z}.$$ For $v=(0,1)$, now we have $G cdot v =(0,1)+L$. In this illustration, we are able to take $P$ to breathe the interval from $(0,1)$ to $(1,1)$.

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