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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Yes. Surface groups are Hopfian. More generally, all residually finite groups are Hopfian  see Theorem IV.4.10 in Lyndon and Schupp's book "Combinatorial Group Theory".

$\begingroup$ I knew I could count on you. Should I just keep trying to prove it myself or is there a reference? $\endgroup$ – Lucas Culler Apr 28 '10 at 3:54

$\begingroup$ I gave a reference for a more general statement  there's probably a more direct proof for surface groups, but the more general statement is extremely useful. For the residual finiteness of surface groups, see Hempel's beautiful 1page paper "Residual finiteness of surface groups". $\endgroup$ – Andy Putman Apr 28 '10 at 3:57

$\begingroup$ And I should also point out that the proof in Lyndon and Schupp I referred to above is only one paragraph long and mostly selfcontained. $\endgroup$ – Andy Putman Apr 28 '10 at 3:58

3$\begingroup$ Not to nitpick, but all finitely generated residually finite groups are Hopfian. $\endgroup$ – Steve D Apr 28 '10 at 4:31

5$\begingroup$ 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. $\endgroup$ – HJRW Apr 28 '10 at 17:19