Homomorphic image of the modular group that has the infinite order.

Question

Let $G$ be a group generated by $A$ and $B$ such that $A^2=B^3=1$ and $AB$ has infinite order. It seems that $G$ should be equal to the modular group $C_2\ast C_3$. Is it true? If yes, how to show this?

Answer 1

I shouldn't really be answering this, because the question is missing context, but it is not true.

For example $ab$ has infinite order in the group $\langle a,b \mid a^2=b^3 = [a,b]^3=1 \rangle$.