# Defining relations of mapping class group for genus 2 closed surface

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 $\Gamma_2$?

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 $2.$
• It seems the section 6 is about $g\geq 3$. – Xiao-Gang Wen May 21 '15 at 22:15
• @Xiao-GangWen true, I was careless in reading. For $g=2$ use the Birman-Hilden (really Bergau-Mennike) presentation instead of Wajnryb's, as now stated. – Igor Rivin May 21 '15 at 22:56