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?
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".
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6 years ago