Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F\_q$ the unique field with $q$ elements.


What is known about the structure and properties of $\mathrm{SL}\_2(F\_q[t])$ as opposed to those of $\mathrm{SL}\_2(\mathbb{Z})$? What do they have in common and what not?


Is this group virtually free as well?


Where can I read more about this group?



Special linear groups over function fields
One important thing that is known about the structure of $SL\_2(\mathbb{F}\_q[t])$ is Nagao's theorem: for any field $k$, in particular $k=\mathbb{F}\_q$, there is an amalgam decomposition
$$
SL\_2(k[t])\cong SL\_2(k)\ast\_{B(k)}B(k[t]),
$$
where $B$ denotes upper triangular matrices. A good reference for statements about the structure of $SL\_2(\mathbb{F}\_q[t])$ would be Serre's book on trees. For instance, Nagao's theorem can be proved using the action of $SL\_2(\mathbb{F}\_q[t])$ on the associated Bruhat-Tits tree. 


There are differences between the number-field situation and the function field situation, in particular where finiteness properties are concerned. While arithmetic groups in the number field situation do have good finiteness properties, $SL\_2(\mathbb{F}\_q[t])$ fails to be finitely generated - because of the infinite amalgam summand $B(\mathbb{F}\_q[t])$. I think the group also fails to be virtually free for the same reason. (But the reason for this difference to the number field situation seems all concentrated at the prime $p$, somehow.)


