Co-Hopfian group

Examples and non-examples

• Every finite group G is co-Hopfian.
• The infinite cyclic group ${\displaystyle \mathbb{Z}}$ is not co-Hopfian since ${\displaystyle f:\mathbb{Z}\to \mathbb{Z},f(n)=2n}$ is an injective but non-surjective homomorphism.
• The additive group of real numbers ${\displaystyle \mathbb{R}}$ is not co-Hopfian, since ${\displaystyle \mathbb{R}}$ is an infinite-dimensional vector space over ${\displaystyle \mathbb{Q}}$ and therefore, as a group ${\displaystyle \mathbb{R}\cong \mathbb{R}\times \mathbb{R}}$.
• The additive group of rational numbers ${\displaystyle \mathbb{Q}}$ and the quotient group ${\displaystyle \mathbb{Q}/\mathbb{Z}}$ are co-Hopfian.
• The multiplicative group ${\displaystyle {\mathbb{Q}}^{\ast }}$ of nonzero rational numbers is not co-Hopfian, since the map ${\displaystyle {\mathbb{Q}}^{\ast }\to {\mathbb{Q}}^{\ast },q\mapsto \mathrm{sign}(q)q^2}$ is an injective but non-surjective homomorphism. In the same way, the group ${\displaystyle {\mathbb{Q}}_{+}^{\ast }}$ of positive rational numbers is not co-Hopfian.
• The multiplicative group ${\displaystyle {\mathbb{C}}^{\ast }}$ of nonzero complex numbers is not co-Hopfian.
• For every ${\displaystyle n\ge 1}$ the free abelian group ${\displaystyle {\mathbb{Z}}^n}$ is not co-Hopfian.
• For every ${\displaystyle n\ge 1}$ the free group ${\displaystyle F_n}$ 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 ${\displaystyle BS(1,m)}$, where ${\displaystyle m\ge 1}$, 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 ${\displaystyle SL(n,\mathbb{Z})}$ for ${\displaystyle n\ge 3}$.
• 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(F_n) (where n>2) is co-Hopfian.
• Delzant and Polyagailo gave a characterization of co-Hopficity for geometrically finite Kleinian groups of isometries of ${\displaystyle {\mathbb{H}}^n}$ without 2-torsion.
• A right-angled Artin group ${\displaystyle A(\mathrm{\Gamma })}$ (where ${\displaystyle \mathrm{\Gamma }}$ is a finite nonempty graph) is not co-Hopfian; sending every standard generator of ${\displaystyle A(\mathrm{\Gamma })}$ to a power ${\displaystyle >1}$ defines and endomorphism of ${\displaystyle A(\mathrm{\Gamma })}$ 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 ${\displaystyle \phi :G\to G}$ is an injective but non-surjective endomorphism of G then either ${\displaystyle {\phi }^k(G)}$ 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 ${\displaystyle \phi :G\to G}$ is an injective endomorphism whose image has finite index in G then ${\displaystyle \phi (G)=G}$. For example, for ${\displaystyle n\ge 2}$ the free group ${\displaystyle F_n}$ 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 ${\displaystyle \phi :G\to G}$ such that ${\displaystyle \bigcap _{n=1}^{\mathrm{\infty }}{\phi }^n(G)=\{1\}}$.
• In coarse geometry, a metric space X is called quasi-isometrically co-Hopf if every quasi-isometric embedding ${\displaystyle f:X\to X}$ is coarsely surjective (that is, is a quasi-isometry). Similarly, X is called coarsely co-Hopf if every coarse embedding ${\displaystyle f:X\to X}$ is coarsely surjective.
• In metric geometry, a metric space K is called quasisymmetrically co-Hopf if every quasisymmetric embedding ${\displaystyle K\to K}$ is onto.