TITLE: A finite generating set for the group $SL_{2} (\mathbb{F}_{2} [t , t^{-1} ] )$
QUESTION [8 upvotes]: $SL_{2} ( \mathbb{F}_{2} [t , t^{-1} ])$ is defined to be the group of $2 \times 2$ matrices with determinant one with entries polynomials in $t$ and $t^{-1}$ with coefficients in the field of two elements, $\mathbb{F}_{2}$. 
By the work of Stuhler (Stuhler, U., Zur Frage der endlichen Prasentierbarkeit gewisser arithmetischer Gruppen im Funktiönenkorperfall, Math. Ann. 224 (1976), 217–232.), it is known that $SL_{2} (\mathbb{F}_{2} [t , t^{-1} ])$ is finitely generated. Is any finite generating set known?

REPLY [10 votes]: I guess it was known earlier (the paper you mention deals with finite presentability, a harder issue).
Denoting $(E_{ij})$ the basis of the space of matrices, $e_{ij}(t)=I+tE_{ij}$, $d_{ij}(a)=aE_{ii}+a^{-1}e_{jj}$, the group $\mathrm{SL}_2(\mathbf{F}_p[t,t^{-1}])$ for $p$ prime is generated by $\{e_{12}(1),e_{12}(t),e_{21}(1),e_{21}(t),d_{12}(t)\}$.
Indeed, $\mathbf{F}_p[t,t^{-1}]$ being a Euclidean ring, $\mathrm{SL}_2(\mathbf{F}_p[t,t^{-1}])$ is generated by elementary matrices. And using conjugation by $d_{12}(t)$ and the four given elementary generators, one obtains all other basis elements among elementary matrices.
(It's not hard to check that $e_{21}(t)$ is redundant in this generating subset, and also that no two of these 5 elements generate the group.)