TITLE: Self-homomorphisms of surface groups
QUESTION [10 upvotes]: 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?

REPLY [10 votes]: 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".