We know that mapping class group (MCG) $\Gamma\_1$ for genus 1 closed surface
is generated by two elements: $U$ of order 6 and $S$ of order 4.
There is a defining relation that totally fixed the MCG $\Gamma\_1$:
$U^3=S^2$. (Is this correct?)


In arXiv:math/0309299, Korkmaz showed that mapping class group $\Gamma\_2$ for genus 2 closed surface is also generated by two elements: $U\_1=A\_4A\_3A\_2A\_1$ of order 10 and $U\_2=A\_5A\_4A\_3A\_2A\_1$ of order 6.


**Do we know the defining relations among the two generators $U\_1$ and $U\_2$ that will allow us to define $\Gamma\_2$?**



Defining relations of mapping class group for genus 2 closed surface
In Section 6 of the Korkmaz paper he uses the Wajnryb presentation to derive a presentation on his two generators, for genus $g>2.$ For genus $2$ one can use the exact same method, but applied to Birman-Hilden's (1971) presentation of the mapping class group of genus $2.$


