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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 is an injective group homomorphism then is surjective, that is .

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 (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 , where , 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 L<sup>2</sup>-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 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 (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<sub>n</sub>) (where n>2) 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 (where is a finite nonempty graph) 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.
  • 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 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 (that is, is a quasi-isometry). 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.

See also

References

Further reading