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 $n\geq 1$ the free abelian group $\mathbb {Z} ^{n}$ is not co-Hopfian.
• For every $n\geq 1$ 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 $m\geq 1$, 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 $n\geq 3$.
• 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 $\mathbb {H} ^{n}$ 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 $n\geq 2$ 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.

See also

• Hopfian object

Further reading

• K. Varadarajan, Hopfian and co-Hopfian Objects, Publicacions Matemàtiques 36 (1992), no. 1, pp. 293–317