Hopfian and Co-Hopfian groups (examples) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:42:10Z http://mathoverflow.net/feeds/question/15115 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples Hopfian and Co-Hopfian groups (examples) Portland 2010-02-12T15:13:53Z 2011-03-28T10:03:45Z <p>Hi, I'm looking for examples of groups that are both Hopfian and Co-Hopfian. I have a non trivial (and beautiful, at least to me) example: $\mathrm{SL}(n,\mathbb{Z})$ (with $n>2$).</p> <p>Do you know others (non trivial)?</p> <p>Thank you.</p> http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples/15119#15119 Answer by Richard Kent for Hopfian and Co-Hopfian groups (examples) Richard Kent 2010-02-12T15:29:17Z 2010-02-12T15:29:17Z <p>Torsion-free $\delta$-hyperbolic groups are hopfian, and it's a theorem of Sela that one-ended torsion free hyperbolic groups are co-hopfian (Z. Sela. Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II. Geom. Funct. Anal., 7(3):561–593, 1997.).</p> http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples/15122#15122 Answer by Johannes Hahn for Hopfian and Co-Hopfian groups (examples) Johannes Hahn 2010-02-12T15:32:43Z 2010-02-12T15:32:43Z <p>$\mathbb{Q}$ is a classical example.</p> http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples/15131#15131 Answer by Igor Belegradek for Hopfian and Co-Hopfian groups (examples) Igor Belegradek 2010-02-12T17:00:51Z 2010-02-12T17:00:51Z <p>Before going to examples, here are some general comments:</p> <p>a) Proving that a finitely generated group is Hopfian is usually pretty hard unless the group is residually finite e.g. finitely generated subgroups of $GL(n,\mathbb R)$ are residually finite, hence Hopfian by an old result of Mal'cev. </p> <p>b) A common method in proving that a group $G$ is co-Hopfian is to use an invariant of $G$ that is multiplicative under passing to finite index subgroups. If $G$ has a nonzero such invariant, then $G$ has no finite index subgroups isomorphic to itself. For example, if $G$ is the fundamental group of a finite aspherical CW-complex of nonzero Euler characteristic, then $G$ has no finite index subgroups isomorphic to itself. </p> <p>c) If $G$ is the fundamental group of a closed aspherical manifold, then $G$ has no infinite index subgroups isomorphic to $G$ (look at top-dimensional homology). </p> <p>d) Euler characteristic, signature, $L^2$-Betti numbers, simplicial volume are are multiplicative under finite covers of closed aspherical manifolds, so if $G$ is the fundamental group of a closed aspherical manifold with say nonzero signature, then $G$ is co-Hopfian.</p> <p>Here are some specific examples to add to Richard's example of one-ended torsion free hyperbolic groups. All of the following groups are linear, hence residually finite, hence Hopfian.</p> <ol> <li><p>The fundamental groups of closed locally symmetric spaces of nonpositive curvature without local flat factors are co-Hopfian because they have nonzero simplicial volume thanks to a result of Lafont-Schmidt.</p></li> <li><p>If memory serves me, it is possible to figure out which geometric 3-manifold groups are co-Hopfian. For example, the $SL_2(\mathbb R)$-Seifert fibered spaces have a certain invariant detecting volume of the base $2$-orbifold which is multiplicative under finite covers. Check papers of Pierre Derbez in arXiv.</p></li> <li><p>Fundamental groups of some nilmanifolds are co-Hopfian, see my paper <a href="http://arxiv.org/abs/math/0302220" rel="nofollow">here.</a></p></li> <li><p>Delzant-Potyagalo classified co-Hopfian Kleinian groups (in real hyperbolic space of any dimension). See <a href="http://www-gat.univ-lille1.fr/~potyag/preprint/endom.ps" rel="nofollow">here.</a> </p></li> </ol> http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples/15154#15154 Answer by Dan Margalit for Hopfian and Co-Hopfian groups (examples) Dan Margalit 2010-02-12T23:45:29Z 2010-02-13T00:20:37Z <p>Mapping class groups of closed surfaces are both Hopfian and co-Hopfian (the former follows from residual finiteness, and the latter is due to Ivanov-McCarthy).</p> <p>Out(F_n) also has both properties (residual finiteness and a theorem of Farb-Handel).</p> http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples/59728#59728 Answer by Colin Reid for Hopfian and Co-Hopfian groups (examples) Colin Reid 2011-03-27T15:08:41Z 2011-03-27T15:08:41Z <p>I think compact $p$-adic analytic groups that have no abelian normal subgroups are Hopfian and co-Hopfian as topological groups, but I haven't seen this explicitly stated anywhere.</p> http://mathoverflow.net/questions/15115/hopfian-and-co-hopfian-groups-examples/59759#59759 Answer by Yiftach Barnea for Hopfian and Co-Hopfian groups (examples) Yiftach Barnea 2011-03-27T20:02:46Z 2011-03-28T10:03:45Z <p>Every open subgroup of the Nottingham group (p>3) is both Hopfian and co-Hopfian. On the one hand, the Nottingham group is hereditarily just infinite. So all it open subgroup are just infinite, that is their proper quotients are finite. On the other hand, Mikhail Ershov showed that the commensurator of the Nottingham group (p>3) is its automorphism group. So if two open subgroups are isomorphic the isomorphism extends to an automorphism of the Nottingham group. In particular, the indices in the Nottingham group of such subgroups are the same. Thus, if one is a subgroup of the other, they are equal.</p> <p><strong>EDIT</strong>: I should be a bit more careful. The fact that the commensurator of the Nottingham group is its automorphism group, does not say that the isomorphism extends to an automorphism, but that the isomorphism restricted to an open subgroup extends to an automorphism. This is good enough for the claim above. It is also may be that Mikhail actually proved the stronger claim that I made (I am not 100% sure).</p> <p><strong>EDIT2</strong>: I have got confused about the definition of co-Hopfian. This argument shows that the Nottingham group is finite co-Hopfian. It is not true that it is co-Hopfian from results of Rachel Camina (and also a paper of Fesenko). </p>