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? 



Self-homomorphisms of surface groups
Yes. Surface groups are [Hopfian](http://en.wikipedia.org/wiki/Hopfian_group). More generally, all [residually finite](http://en.wikipedia.org/wiki/Residually_finite) groups are Hopfian -- see Theorem IV.4.10 in Lyndon and Schupp's book "Combinatorial Group Theory".


