Co-Hopfian group

Formal definition

Examples and non-examples

• Every finite group G is co-Hopfian.
• The infinite cyclic group is not co-Hopfian since is an injective but non-surjective homomorphism.
• The additive group of real numbers is not co-Hopfian, since is an infinite-dimensional vector space over and therefore, as a group.
• The additive group of rational numbers and the quotient group are co-Hopfian.
• The multiplicative group of nonzero rational numbers is not co-Hopfian, since the map is an injective but non-surjective homomorphism. In the same way, the group of positive rational numbers is not co-Hopfian.
• The multiplicative group of nonzero complex numbers is not co-Hopfian.
• For every the free abelian group is not co-Hopfian.
• For every the free group is not co-Hopfian.
• There exists a finitely generated non-elementary 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, where, are not co-Hopfian.
• If G is the fundamental group of a closed aspherical manifold with nonzero Euler characteristic, 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 for.
• 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 of pinched negative curvature then G is co-Hopfian.
• The mapping class group of a closed hyperbolic surface is co-Hopfian.
• The group Out is co-Hopfian.
• Delzant and Polyagailo gave a characterization of co-Hopficity for geometrically finite Kleinian groups of isometries of without 2-torsion.
• A right-angled Artin group is not co-Hopfian; sending every standard generator of to a power defines and endomorphism of 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 is an injective but non-surjective endomorphism of G then either 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.
• Thomposon group F is not co-Hopfian.
• There exists a finitely generated group G which is not co-Hopfian but has Kazhdan's property.
• 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 is an injective endomorphism whose image has finite index in G then. For example, for the free group 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 such that.
• In coarse geometry, a metric space X is called quasi-isometrically co-Hopf if every quasi-isometric embedding is coarsely surjective. Similarly, X is called coarsely co-Hopf if every coarse embedding is coarsely surjective.
• In metric geometry, a metric space K is called quasisymmetrically co-Hopf if every quasisymmetric embedding is onto.