I'm trying to show that $SL_2(\mathbb{Z})\cong G=\langle a,b|a^4=1,a^2b^{-3}=1\rangle$.
For this, I defined a homomorphism $f$ from $G$ into $SL_2(\mathbb{Z})$ by $$f(a)=\begin{bmatrix}0& -1\\1& 0\end{bmatrix},\ \ f(b)=\begin{bmatrix}0& -1\\1& 1\end{bmatrix}$$
I proved that it is surjective, but have no idea to prove it is injective.
My text book suggests every word $g \in G$ can be written as $g=a^{2i}a^{p_1}b^{q_1}a^{p_2}b^{q_2}\cdots$ where $p_j=0,1$ and $q_j=0,1,2$, so one can use the fact that every word is equivalent to exactly one reduced word for $\mathbb{Z_2}*\mathbb{Z_3}$.
I showed the words in $G$ can be written as the above, but don't know how to apply the uniqueness of a reduced word to prove the injectivity.
Any help will be appreciated.