I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not clear why they would do that?
I thought about it and can see how we can identify unimodular lattices with elements of $SL(2, \mathbb{Z})$. Why would we then want to quotient by that space?
A basis $B$ of a lattice can be identified with a matrix $[B]$ whose columns are the vectors of $B$. Two bases $B_1$ and $B_2$ span the same lattice if and only if they are equivalent up to multiplication by a unimodular matrix (i.e. if there exists $U \in \mathbb{Z}^{n \times n}$ such that $[B_2] = U[B_1]$). As such, we have to mod by $SL(2, \mathbb{Z})$ if we want equivalence classes of matrices generating the same lattices.