I am trying to understand a proof given of an isomorphism between an infinite and finite presentation of Thompson's group F in the following paper by Cannon, Floyd and Parry.
I'm referring to Theorem 3.1 on page 7.
So, I understand how to construct the surjective homomorphism (I'll just call it $\phi$) mentioned at the beginning. It's the steps after this that I don't quite understand. How does:
$(i)$ Showing that the defining relations of $F_1$ are contained in the kernel of $\phi$
$(ii)$ Showing that there exists a homomorphism the other way (from $F_2$ to $F_1$)
Prove the result?
To show $\phi$ is injective surely you have to show that the defining relations of $F_1$ are equal to ker($\phi$)? (since then $\phi$ has trivial kernel)
Also, since $\phi$ is defined by $\phi(A) = X_0$ and $\phi(B) = X_1$, does showing there exists a homomorphism the other way that maps $X_0$ to $A$ and $X_1$ to $B$ not prove that $\phi$ has an inverse, and so must be an isomorphism anyway? If so, why bother to show $(i)$?
Thanks in advance for any help