Co-Hopfian group

In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.

Formal definition
A group G is called co-Hopfian if whenever $$\varphi:G\to G$$ is an injective group homomorphism then $$\varphi$$ is surjective, that is $$\varphi(G)=G$$.

Examples and non-examples

 * Every finite group G is co-Hopfian.
 * The infinite cyclic group $$\mathbb Z$$ is not co-Hopfian since $$f:\mathbb Z\to \mathbb Z, f(n)=2n$$ is an injective but non-surjective homomorphism.
 * The additive group of real numbers $$\mathbb R$$ is not co-Hopfian, since $$\mathbb R$$ is an infinite-dimensional vector space over $$\mathbb Q$$ and therefore, as a group $$\mathbb R\cong \mathbb R\times \mathbb R$$.
 * The additive group of rational numbers $$\mathbb Q$$ and the quotient group $$\mathbb Q/\mathbb Z$$ are co-Hopfian.
 * The multiplicative group $$\mathbb Q^\ast$$ of nonzero rational numbers is not co-Hopfian, since the map $$\mathbb Q^\ast\to\mathbb Q^\ast, q\mapsto \operatorname{sign}(q)\,q^2$$ is an injective but non-surjective homomorphism. In the same way, the group $$\mathbb Q^{\ast}_+$$ of positive rational numbers is not co-Hopfian.
 * The multiplicative group $$\mathbb C^\ast$$ of nonzero complex numbers is not co-Hopfian.
 * For every $$n\ge 1$$ the free abelian group $$\mathbb Z^n$$ is not co-Hopfian.
 * For every $$n\ge 1$$ the free group $$F_n$$ is not co-Hopfian.
 * There exists a finitely generated non-elementary (that is, not virtually cyclic) virtually free group which is co-Hopfian. Thus a subgroup of finite index in a finitely generated co-Hopfian group need not be co-Hopfian, and being co-Hopfian is not a quasi-isometry invariant for finitely generated groups.
 * Baumslag–Solitar groups $$BS(1,m)$$, where $$m\ge 1$$, are not co-Hopfian.
 * If G is the fundamental group of a closed aspherical manifold with nonzero Euler characteristic (or with nonzero simplicial volume or nonzero L2-Betti number), then G is co-Hopfian.
 * If G is the fundamental group of a closed connected oriented irreducible 3-manifold M then G is co-Hopfian if and only if no finite cover of M is a torus bundle over the circle or the product of a circle and a closed surface.
 * If G is an irreducible lattice in a real semi-simple Lie group and G is not a virtually free group then G is co-Hopfian. E.g. this fact applies to the group $$SL(n,\mathbb Z)$$ for $$n\ge 3$$.
 * If G is a one-ended torsion-free word-hyperbolic group then G is co-Hopfian, by a result of Sela.
 * If G is the fundamental group of a complete finite volume smooth Riemannian n-manifold (where n > 2) of pinched negative curvature then G is co-Hopfian.
 * The mapping class group of a closed hyperbolic surface is co-Hopfian.
 * The group Out(Fn) (where n>2) is co-Hopfian.
 * Delzant and Polyagailo gave a characterization of co-Hopficity for geometrically finite Kleinian groups of isometries of $$\mathbb H^n$$ without 2-torsion.
 * A right-angled Artin group $$A(\Gamma)$$ (where $$\Gamma$$ is a finite nonempty graph) is not co-Hopfian; sending every standard generator of $$A(\Gamma)$$ to a power $$>1$$ defines and endomorphism of $$A(\Gamma)$$  which is injective but not surjective.
 * A finitely generated torsion-free nilpotent group G may be either co-Hopfian or not co-Hopfian, depending on the properties of its associated rational Lie algebra.
 * If G is a relatively hyperbolic group and $$\varphi:G\to G$$ is an injective but non-surjective endomorphism of G then either $$\varphi^k(G)$$ is parabolic for some k >1 or G splits over a virtually cyclic or a parabolic subgroup.
 * Grigorchuk group G of intermediate growth is not co-Hopfian.
 * Thompson group F is not co-Hopfian.
 * There exists a finitely generated group G which is not co-Hopfian but has Kazhdan's property (T).
 * If G is Higman's universal finitely presented group then G is not co-Hopfian, and G cannot be embedded in a finitely generated recursively presented co-Hopfian group.

Generalizations and related notions

 * A group G is called finitely co-Hopfian if whenever $$\varphi:G\to G$$ is an injective endomorphism whose image has finite index in G then $$\varphi(G)=G$$. For example, for $$n\ge 2$$ the free group $$F_n$$ is not co-Hopfian but it is finitely co-Hopfian.
 * A finitely generated group G is called scale-invariant if there exists a nested sequence of subgroups of finite index of G, each isomorphic to G, and whose intersection is a finite group.
 * A group G is called dis-cohopfian if there exists an injective endomorphism $$\varphi:G\to G$$ such that $$\bigcap_{n=1}^\infty \varphi^n(G)=\{1\}$$.
 * In coarse geometry, a metric space X is called quasi-isometrically co-Hopf if every quasi-isometric embedding $$f:X\to X$$ is coarsely surjective (that is, is a quasi-isometry). Similarly, X is called coarsely co-Hopf if every coarse embedding $$f:X\to X$$ is coarsely surjective.
 * In metric geometry, a metric space K is called quasisymmetrically co-Hopf if every quasisymmetric embedding $$K\to K$$ is onto.