# Self-homomorphisms of surface groups

Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi_1(X) \to \pi_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?

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The proof is easy enough to write in a comment. Suppose g is in the kernel of a surjection $\phi:G\to G$ and let $q: G\to Q$ be a finite quotient with $q(g)\neq 1$. Then it's easy to see that $q\circ\phi^n$ are all distinct maps $G\to Q$. But there are only finitely many maps from a finitely generated group to a fixed finite group. –  HJRW Apr 28 '10 at 17:19