Meaning of “modulo the identity matrix”

Solution 1:

You asked about the quote

the matrices in $SL(2,R)$ modulo plus or minus the identity matrix.

In the context of $\,PSL(2,R),\,$ a matrix $\,A={a\, b\choose c\,d}\,$ is mapped to the linear fractional transformation $\,z\mapsto \frac{az+b}{cz+d}\,$ which is a homomorphism of groups as in Wikipedia SL(2,R). The matrices $\,A\,$ and $\,-A\,$ both map to the same transformation (in general, any scalar multiple of $\,A\,$ maps to the same transformation). The identity matrix and its negative form a normal subgroup of $\,SL(2,R)\,$ whose quotient is $\,PSL(2,R)\,$ where, indeed, $\,A\,$ and $\,-A\,$ are identified in the quotient.

Of course, if $\,R=\mathbb{Z}\,$ then this is the only identification of matrices when passing to linear transformations. For other rings, there may be more identifications based on units in the ring $\,R\,$ as indicated in the Wikipedia article.

You asked:

2. Is this homomorphism the squaring function? If not, what's the homomorphism?

The "squaring" function is not a homomorphism.

3. The 2x2 matrices ([3,1],[-7,-2]) and ([-3,-1],[7,2]) are both in 𝑆𝐿(2,𝑅). But only one is in 𝑃𝑆𝐿(2,𝑅), right? Which one and why?

The elements of 𝑃𝑆𝐿(2,𝑅) are cosets (or equivalence classes) and each coset consists of a matrix and its negative, so both are in the same coset of 𝑃𝑆𝐿(2,𝑅).