Every surjective homomorphism from a fundamental group of a connected and compact manifold to itself is an isomorphism

Question

I can't understand why every surjective homomorphism from a fundamental group of a connected and compact manifold to itself is an isomorphism? Is there any classification fore such groups?

Answer 1

Such groups $G$ are called Hopfian. Certainly not all finitely-generated groups are Hopfian. For a simple counterexample consider the groups $BS(m,n) = \langle b,s\mid s^{-1}b^ms = b^n\rangle$ with $2$ generators and $1$ relation constructed by Baumslag and Solitar for certain $(m,n)$, e.g., $(m,n)=(2,3)$. Actually $BS(m,n)$ is Hopfian if and only if $m,n$ equals $\pm 1$, or $m$ and $n$ have the same set of prime divisors.

By Michal's remark these groups arise as the fundamental group of a connected compact $4$-manifold. So the claim in the title is not true.