I am trying to understand the below proof better:
2026-03-27 08:47:02.1774601222
Limit groups are CSA
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I have to say I don't understand your question 1. Surely you know what "injective" means?
Actually, I think there may be a small typo in the proof, which is perhaps what is causing your confusion for question 2. The hypothesis $h\in gHg^{-1}\cap H$ means that $h\in H$ and $g^{-1}hg\in H$ (not $ghg^{-1}$, as written). Therefore, since $H$ is abelian, $h$ and $g^{-1}hg$ commute, ie $[h,g^{-1}hg]=1$.
So far, this is all completely elementary. Question 3 just needs a very basic fact about free groups, namely that the centralizer of any non-trivial element is cyclic. The identity $f([h,g^{-1}hg])=1$ implies that $f(h)$ and $f(g^{-1}hg)$ commute, so both live in the centralizer of $f(h)$, which is a cyclic subgroup.