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Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

To illustrate how such a question is of arithmetic nature (and what a complete answer should look like), here is a partial answer in a specific case. Namely: I specify to $M=\begin{pmatrix}2 & 1 \\...
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1 vote

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

For A fixed $M$, such finite index subgroups may be categorized as follows. Let $W\in GL(2,\mathbb{Z})$ and $b$ a positive integer dividing the lower left element of the matrix $W^{-1} M W$. Then the ...
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3 votes

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

If $v$ is not scalar, the ring $\mathbb Z[M]$ generated by $M$ has rank $2$ over $\mathbb Z$. It is therefore either an order in a number field, a finite-index subring of $\mathbb Z^2$, or $\mathbb Z[\...
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3 votes

The smallest volume possible for a lattice with integer distances?

Note that $\sqrt2\Lambda$ is even, and $\mathrm{vol}(\sqrt2\Lambda)^2=\det(A)$, where $A$ is the Gram matrix of $\sqrt2\Lambda$. Thus $\mathrm{vol}(\Lambda)=2^{-n/2}\sqrt{\det(A)}$, so the smallest ...
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10 votes
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The smallest volume possible for a lattice with integer distances?

After scaling your lattice by $\sqrt{2}$, the Gram matrix has integral entries, so the absolute value of its determinant, being a nonzero integer, is at least $1$. This gives a lower bound of $2^{-n/2}...
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