I'm investigating $SL_2$ over some superset of the products of positive dyadic and ternary rationals $\Bbb Z[\frac16]^+$.
For my application it is useful to have the 2-adic value on hand. To this end the superset I am currently using as a setting is the field $\Bbb Q_2$ , although this is not set in stone. And what is more, it is useful for me to have the 3-adic value on hand, so to this end I am restricting myself to the periodic elements of $\Bbb Q_2$ insofar as is possible.
In order to understand $SL(2,\Bbb Q_2)$ I have the statement for $\Bbb R$:
The group action factors through the quotient $PSL(2, \Bbb R)$ (the 2 × 2 projective special linear group over $\Bbb R$).
Question
What's the 2-adic analogue of this statement - is it any different, and what does it mean?
Background
I don't think this affects the answer that much but it explains the motivation and that may allow a more helpful answer: The function of the well-known Collatz conjecture can be seen as acting on the quotient $\Bbb Z[\frac16]^+/\langle2,3\rangle$, which you can think of as having one 5-rough number as a representative of every element - which you arrive at by dividing out the powers of two and three.
We get a rather elegant representation of the Collatz conjecture within the special linear group over the 2-adic field $SL_2(\Bbb Q_2)$ using the matrix $\begin{pmatrix}1&&2^{\nu_2(x)}\cdot3^{\nu_3(x)-1}\\0&&1\end{pmatrix}$. This is an epimorphism on $\Bbb Q_2$ (assuming some means of assigning powers of $3$ to the aperiodic elements).
This matrix is just one "choice" of infinitely many possible representations of the Collatz function which respects the rays of the quotient $/\langle2,3\rangle$, but it is the one which sits within $SL_2(\Bbb Q_2)$ and it also happens to be the one for which the level sets of the function accumulate to their image, so it seems to have the better claim to be the "chosen" representative. I'd like to understand more about this and the implications of it factoring through the quotient $PSL(2,\Bbb Q_2)$, if that is indeed the 2-adic equivalent of the statement for $\Bbb R$.