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. [1]

Formal definition

A group G is called co-Hopfian if whenever ${\displaystyle \varphi :G\to G}$ is an injective group homomorphism then ${\displaystyle \varphi }$ is surjective, that is ${\displaystyle \varphi (G)=G}$.[2]

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 ${\displaystyle \mathbb {R} \cong \mathbb {R} \times \mathbb {R} }$.[2]
• The additive group of rational numbers ${\displaystyle \mathbb {Q} }$ and the quotient group ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ are co-Hopfian.[2]
• The multiplicative group ${\displaystyle \mathbb {Q} ^{\ast }}$ of nonzero rational numbers is not co-Hopfian, since the map ${\displaystyle \mathbb {Q} ^{\ast }\to \mathbb {Q} ^{\ast },q\mapsto \operatorname {sign} (q)\,q^{2}}$ is an injective but non-surjective homomorphism.[2] 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.[2]
• For every ${\displaystyle n\geq 1}$ the free abelian group ${\displaystyle \mathbb {Z} ^{n}}$ is not co-Hopfian.[2]
• For every ${\displaystyle n\geq 1}$ the free group ${\displaystyle F_{n}}$ is not co-Hopfian.[2]
• 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.[3]
• Baumslag–Solitar groups ${\displaystyle BS(1,m)}$, where ${\displaystyle m\geq 1}$, are not co-Hopfian.[4]
• 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.[5]
• 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.[6]
• 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.[7] E.g. this fact applies to the group ${\displaystyle SL(n,\mathbb {Z} )}$ for ${\displaystyle n\geq 3}$.
• If G is a one-ended torsion-free word-hyperbolic group then G is co-Hopfian, by a result of Sela.[8]
• 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. [9]
• The mapping class group of a closed hyperbolic surface is co-Hopfian.[10]
• The group Out(F_n) (where n>2) is co-Hopfian.[11]
• Delzant and Polyagailo gave a characterization of co-Hopficity for geometrically finite Kleinian groups of isometries of ${\displaystyle \mathbb {H} ^{n}}$ without 2-torsion.[12]
• A right-angled Artin group ${\displaystyle A(\Gamma )}$ (where ${\displaystyle \Gamma }$ is a finite nonempty graph) is not co-Hopfian; sending every standard generator of ${\displaystyle A(\Gamma )}$ to a power ${\displaystyle >1}$ defines and endomorphism of ${\displaystyle A(\Gamma )}$ which is injective but not surjective.[13]
• 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.[5][3]
• If G is a relatively hyperbolic group and ${\displaystyle \varphi :G\to G}$ is an injective but non-surjective endomorphism of G then either ${\displaystyle \varphi ^{k}(G)}$ is parabolic for some k >1 or G splits over a virtually cyclic or a parabolic subgroup.[14]
• Grigorchuk group G of intermediate growth is not co-Hopfian.[15]
• Thomposon group F is not co-Hopfian.[16]
• There exists a finitely generated group G which is not co-Hopfian but has Kazhdan's property (T).[17]
• 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.[18]

Generalizations and related notions

• A group G is called finitely co-Hopfian[19] if whenever ${\displaystyle \varphi :G\to G}$ is an injective endomorphism whose image has finite index in G then ${\displaystyle \varphi (G)=G}$. For example, for ${\displaystyle n\geq 2}$ 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.[4]
• A group G is called dis-cohopfian[3] if there exists an injective endomorphism ${\displaystyle \varphi :G\to G}$ such that ${\displaystyle \bigcap _{n=1}^{\infty }\varphi ^{n}(G)=\{1\}}$.
• In coarse geometry, a metric space X is called quasi-isometrically co-Hopf if every quasi-isometric embedding ${\displaystyle f:X\to X}$ 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. [20]
• In metric geometry, a metric space K is called quasisymmetrically co-Hopf if every quasisymmetric embedding ${\displaystyle K\to K}$ is onto. [21]

See also

• Hopfian object

References

1. Wilhelm Magnus, Abraham Karrass, Donald Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, Reprint of the 1976 second edition, Dover Publications, Inc., Mineola, NY, 2004. ISBN 0-486-43830-9
2. P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6; p. 58
3. Yves 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
4. Volodymyr Nekrashevych, and Gábor Pete, Scale-invariant groups. Groups, Geometry, and Dynamics 5 (2011), no. 1, pp. 139–167
5. Igor Belegradek, On co-Hopfian nilpotent groups. Bulletin of the London Mathematical Society 35 (2003), no. 6, pp. 805–811
6. Shi Cheng Wang, and Ying Qing Wu, Covering invariants and co-Hopficity of 3-manifold groups. Proceedings of the London Mathematical Society 68 (1994), no. 1, pp. 203–224
7. Gopal Prasad Discrete subgroups isomorphic to lattices in semisimple Lie groups. American Journal of Mathematics 98 (1976), no. 1, 241–261
8. Zlil Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II. Geometric and Functional Analysis 7 (1997), no. 3, pp. 561–593
9. I. Belegradek, On Mostow rigidity for variable negative curvature. Topology 41 (2002), no. 2, pp. 341–361
10. Nikolai Ivanov and John McCarthy, On injective homomorphisms between Teichmüller modular groups. I. Inventiones Mathematicae 135 (1999), no. 2, pp. 425–486
11. Benson Farb and Michael Handel, Commensurations of Out(F_n), Publications Mathématiques de l'IHÉS 105 (2007), pp. 1–48
12. Thomas Delzant and Leonid Potyagailo, Endomorphisms of Kleinian groups. Geometric and Functional Analysis 13 (2003), no. 2, pp. 396–436
13. Montserrat Casals-Ruiz, Embeddability and quasi-isometric classification of partially commutative groups. Algebraic and Geometric Topology 16 (2016), no. 1, 597–620
14. Cornelia Druţu and Mark Sapir, Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups. Advances in Mathematics 217 (2008), no. 3, pp. 1313–1367
15. Igor Lysënok, A set of defining relations for the Grigorchuk group. (in Russian) Matematicheskie Zametki 38 (1985), no. 4, 503–516
16. Bronlyn Wassink, Subgroups of R. Thompson's group F that are isomorphic to F. Groups, Complexity, Cryptology 3 (2011), no. 2, 239–256
17. Yann Ollivier, and Daniel Wise, Kazhdan groups with infinite outer automorphism group. Transactions of the American Mathematical Society 359 (2007), no. 5, pp. 1959–1976
18. Charles F. Miller, and Paul Schupp, Embeddings into Hopfian groups. Journal of Algebra 17 (1971), pp. 171–176
19. Martin Bridson, Daniel Groves, Jonathan Hillman, Gaven Martin, Cofinitely Hopfian groups, open mappings and kno complements. Groups, Geometry, and Dynamics 4 (2010), no. 4, pp. 693–707
20. Ilya Kapovich, and Anton Lukyanenko, Quasi-isometric co-Hopficity of non-uniform lattices in rank-one semi-simple Lie groups. Conformal Geometry and Dynamics 16 (2012), pp. 269–282
21. Sergei Merenkov, A Sierpiński carpet with the co-Hopfian property. Inventiones Mathematicae 180 (2010), no. 2, pp. 361–388

Further reading

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