Hopfian and co-Hopfian groups

Question

Let $G$ be a group. We say that $G$ is a Hopfian(Co-Hopfian) group, if every epimorphism(monomorphism) of $G$ if a monomorphism(epimorphism). Can someone give me some applications of Hopfian and co-Hopfian groups in geometry...?

Answer 1

Hopf was originally interested in surfaces, and maps between them. In his original paper$^{\dagger}$ he wrote,

"I believe the problem of listing all classes of maps of the closed orientable surface of genus $g$ onto the closed orientable surface of genus $g$ is interesting, both on account of the connection with function-theoretical [analytic?] questions...as well as from a purely topological viewpoint. This question is only solved for special cases...Apart from these, it is easy to show that the sought-after list is identical to that of all homomorphisms of the fundamental group of the manifold into the fundamental group of the same manifold. However, this group-theoretic problem is most likely no easier to solve than the original geometric one." -- Translated once upon a time by one of my German friends.

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$^{\dagger}$H. Hopf, Beiträge zur Klassifizierung der Flachenabbildungen, J. Reine Angew. Math., 165, pp 225-236, 1931