It's stated in Serre's book "Trees" that $SL_2(\mathbb{Z}[1/p])$ is the amalgamated product of two copies of $SL_2(\mathbb{Z})$ along the subgroup $\Gamma_0(p)$. Unfortunately, it doesn't give the isomorphism.
My question is - what is this isomorphism? In particular, what is a minimal set of generators for $SL_2(\mathbb{Z}[1/p])$?
According to Section 4 of Behr-Mennicke's "A Presentation of the Groups $PSL(2,p)$", a set of generators are:
$$A = \begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}\qquad B = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}\qquad U = \begin{bmatrix}p & 0 \\0 & p^{-1}\end{bmatrix}$$
They even give a full set of relations:
$$B^2 = (AB)^3,\qquad B^4=1,\qquad U^{-1}AU = A^{p^2},\qquad (UB)^2 = B^2$$ where the first two are relations for $SL_2(\mathbb{Z})$.
When $p = 2$ there is an additional relation: $$(UA^2B)^3 = B^2$$