$SL\_{2} ( \mathbb{F}\_{2} [t , t^{-1} ])$ 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? 



A finite generating set for the group $SL_{2} (\mathbb{F}_{2} [t , t^{-1} ] )$
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.)


