I have a group presentation, and I would like to see if it embeds into GL(2, Z). Now, to me this sounds like rep theory, which is something I know little (well, pretty much nothing) about. I'd rather not discuss the group in question, so I suppose my question can be asked in two parts,

How can one tell if a given group embeds into GL(2, Z)?

If a given group does embed, how would we find a representation?

For example, I know that $\displaystyle D_{\infty}=\langle a, b; b^2, bab=a^{-1}\rangle$ embeds in GL(2, Z), and I was wondering how I would work out a representation? (I worked out a representation by finding it as a subgroup of $\displaystyle Aut(\mathbb{Z}^2)$, which is isomorphic to GL(2, Z)...but I think this is overly complicated...the subgroup is the one generated by the two automorphisms $\displaystyle a\mapsto ab, b\mapsto b$ and $\displaystyle a\mapsto a^{-1}, b\mapsto b$, although one can take $\displaystyle a\mapsto a, b\mapsto b^{-1}$ as the second one, and it wouldn't matter...this gives the representation generated by,

$\displaystyle \left( \begin{array}{cc}1 & 1 \\0 & 1 \end{array} \right)$

$\displaystyle \left( \begin{array}{cc}-1 & 0 \\0 & 1 \end{array} \right)$

but, well, surely there is an easier way?...)