Finite Presentation of a subgroup

Question

I have the group $\langle a,b \mid a^3b^3\rangle$ Now I send both $a$ and $b$ to the generator of $\mathbb{Z}/3\mathbb{Z}$. This gives a well-defined homomorphism from our group to $\mathbb{Z}/3\mathbb{Z}$ and I am asked to find a finite presentation of the kernel of this homomorphism. How do I generally tackle these kind of questions?

I know about covering spaces, fundamental groups of spaces, I understood I should take a cell complex and then find a cover corresponding to that subgroup, but I don't get how to do it so far.

Answer 1

Here I get a presentation of the kernel: $$\langle x_1,x_2,x_3,x_4\mid x_2x_3x_4\rangle$$ where \begin{array}l x_1:=a^3,\\ x_2:=ba^{-1},\\ x_3:=aba^{-2},\\ x_4:=a^2b\end{array}