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 $\varphi(G)=G$.P. de la Harpe
''Topics in geometric group theory''.
Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ; p. 58

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 $\mathbb Q^\ast\to\mathbb Q^\ast, q\mapsto \operatorname(q)\,q^2$ is an injective but non-surjective homomorphism. In the same way, the group $\mathbb Q^_+$ 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(''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.Igor Belegradek, ''On co-Hopfian nilpotent groups''. Bulletin of the London Mathematical Society 35 (2003), no. 6, pp. 805–811Yves Cornulier, ''Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups''. Bulletin de la Société Mathématique de France 144 (2016), no. 4, pp. 693–744
*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.Volodymyr Nekrashevych, and Gábor Pete, ''Scale-invariant groups''. Groups, Geometry, and Dynamics 5 (2011), no. 1, pp. 139–167
*A group ''G'' is called dis-cohopfian if there exists an injective endomorphism $\varphi:G\to G$ such that $\bigcap_^\infty \varphi^n(G)=\$.
*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.Sergei Merenkov, ''A Sierpiński carpet with the co-Hopfian property''. Inventiones Mathematicae 180 (2010), no. 2, pp. 361–388

See also

*Hopfian object

References

{{Reflist

Further reading

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

Group theory