$\newcommand{\R}{\mathbf R}$ Let $G$ be the group of homeomorphisms of $\R^2$ generated by $g$ and $h$, where $g(x, y)=(x+1, y)$ and $h(x, y)=(-x, y+1)$.
To show that $G\cong \langle a, b|\ b^{-1}aba\rangle$.
I tried the following:
Define a map $f:\langle a, b\rangle \to G$ which sends $a$ to $g$ and $b$ to $h$. Then it can be checked that $b^{-1}aba$ lies in the kernel of $f$. So $f$ factors through $\langle a, b|\ b^{-1}aba\rangle$ to give a map $\bar f: \langle a, b|\ b^{-1}aba\rangle\to G$.
What I am unable to show is that $\bar f$ is injective.
Also, here we were already given a presentation which we had to show is isomorphic to $G$. If it were not given, then is there a general way to get one?
Thank you.