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.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.

Formal definition

A group G is called co-Hopfian if whenever is an injective group homomorphism then is surjective, that is .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 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 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.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 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.Gopal Prasad Discrete subgroups isomorphic to lattices in semisimple Lie groups. American Journal of Mathematics 98 (1976), no. 1, 241–261 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.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 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.I. Belegradek, On Mostow rigidity for variable negative curvature. Topology 41 (2002), no. 2, pp. 341–361 The mapping class group of a closed hyperbolic surface is co-Hopfian.Nikolai Ivanov and John McCarthy, On injective homomorphisms between Teichmüller modular groups. I. Inventiones Mathematicae 135 (1999), no. 2, pp. 425–486 The group Out(Fn) (where n>2) is co-Hopfian.Benson Farb and Michael Handel, Commensurations of Out(Fn), Publications Mathématiques de l'IHÉS 105 (2007), pp. 1–48 Delzant and Polyagailo gave a characterization of co-Hopficity for geometrically finite Kleinian groups of isometries of without 2-torsion.Thomas Delzant and Leonid Potyagailo, Endomorphisms of Kleinian groups. Geometric and Functional Analysis 13 (2003), no. 2, pp. 396–436 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.Montserrat Casals-Ruiz, Embeddability and quasi-isometric classification of partially commutative groups. Algebraic and Geometric Topology 16 (2016), no. 1, 597–620 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 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.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 Grigorchuk group G of intermediate growth is not co-Hopfian.Igor Lysënok, A set of defining relations for the Grigorchuk group. Matematicheskie Zametki 38 (1985), no. 4, 503–516 Thompson group F is not co-Hopfian.Bronlyn Wassink, Subgroups of R. Thompson's group F that are isomorphic to F. Groups, Complexity, Cryptology 3 (2011), no. 2, 239–256 There exists a finitely generated group G which is not co-Hopfian but has Kazhdan's property (T).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 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.Charles F. Miller, and Paul Schupp, Embeddings into Hopfian groups. Journal of Algebra 17 (1971), pp. 171–176

Generalizations and related notions

A group G is called finitely co-HopfianMartin Bridson, Daniel Groves, Jonathan Hillman, Gaven Martin, Cofinitely Hopfian groups, open mappings and knot complements. Groups, Geometry, and Dynamics 4 (2010), no. 4, pp. 693–707 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.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 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.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 In metric geometry, a metric space K is called quasisymmetrically co-Hopf if every quasisymmetric embedding is onto.Sergei Merenkov, A Sierpiński carpet with the co-Hopfian property. Inventiones Mathematicae 180 (2010), no. 2, pp. 361–388

Co-Hopfian group

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