TITLE: Hopfian and Co-Hopfian groups (examples)
QUESTION [9 upvotes]: Hi,
I'm looking for examples of groups that are both Hopfian and Co-Hopfian. I have a non trivial (and beautiful, at least to me) example: $\mathrm{SL}(n,\mathbb{Z})$ (with $n>2$).
Do you know others (non trivial)?
Thank you.

REPLY [2 votes]: Torsion-free finitely generated nilpotent groups are Hopfian. Although the easiest (nontrivial) ones (such as abelian ones, Heisenberg, those occurring in unipotent radicals of parabolics of reductive groups...) are not cohopfian, many are cohopfian too.
An iff condition is that the Malcev completion (say, the complex one, but it works equally with the rational or real one) has a nontrivial non-negative grading. See my paper. The initial observation showing their existence, with a sufficient (but not necessary) condition, namely that the Malcev completion is characteristically nilpotent, appears in I. Belegradek arXiv link and earlier in D. Segal's MR review of Smith, Geoff C. Compressibility in nilpotent groups.
Bull. London Math. Soc. 17 (1985), no. 5, 453–457. Quoth:

It is perhaps not so well known that finitely generated torsion-free nilpotent groups need not be compressible: this follows from J. L. Dyer's construction of a nilpotent Lie algebra all of whose automorphisms are unipotent [Bull. Amer. Math. Soc. 76 (1970), 52–56.

(In turn, Dyer 1970 is not the original reference for the latter fact, but Dixmier, J.; Lister, W., Derivations of nilpotent Lie algebras.
Proc. Amer. Math. Soc. 8 (1957), 155–158.)