Timeline for Hopfian and Co-Hopfian groups (examples)
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2021 at 9:54 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Mar 28, 2011 at 12:59 | comment | added | Yiftach Barnea | Igor, thanks for the clarification. | |
| Mar 28, 2011 at 11:59 | comment | added | Igor Belegradek | @Yiftach, I did not want to use the term "finitely co-Hopfian", but checking this property is part of the task of checking whether the group is co-Hopf, and in b) I explain a most common way to do so. The two properties are equivalent for the fundamental groups of closed aspherical $n$-manifolds (as the cohomological dimension of the group is $n$, while any infinite index subgroup has cohomological dimension $<n$ being the fundamental group of a noncompact aspherical $n$-manifold). On the other hand a (finitely generated nonabelian) free group is finitely co-Hopf but not co-Hopf. | |
| Mar 28, 2011 at 9:59 | comment | added | Yiftach Barnea | Igor, your answer confused me. What is the definition of co-Hopfian? Part b of your answer implies that it means not to be isomorphic to a subgroup of finite index. However, that is not the definition in Planenmath planetmath.org/encyclopedia/HopfianGroup.html. Here journals.cambridge.org/… it is called finite co-Hopfian. | |
| Feb 12, 2010 at 17:00 | history | answered | Igor Belegradek | CC BY-SA 2.5 |