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Does there exist a finitely presented, torsion-free group $G$ which has conjugacy classes of finite size greater than one?

This condition came up in a research project, and we would like to rule out the existence of such examples.

Edit: Sorry for the confusion. I meant to ask for an example of such $G$ such that some conjugacy class is of finite size greater than one. Colin Reid's comment shows that the Klein bottle group is an example.

Let me ask a follow-up: If $G$ is f.p. torsion-free and $G/Z(G)=\operatorname{Inn}(G)$ is torsion-free, then is every conjugacy class of $G$ either central or infinite?

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    $\begingroup$ Do you want all conjugacy classes to be finite, or just for there to exist a finite noncentral conjugacy class? As Mark Sapir says, the former can't happen, but the latter can easily happen, e.g. <x,y|xyx^{-1} = y^{-1}>. $\endgroup$
    – Colin Reid
    Jul 17, 2017 at 9:47
  • $\begingroup$ As Colin Reid says, this question has a very high index of ambiguity! All classes or some classes? $\endgroup$
    – Derek Holt
    Jul 17, 2017 at 10:18
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    $\begingroup$ For your follow-up question, take a semidirect product ${\mathbb Z}^3 \rtimes_\phi {\mathbb Z}$, where $\phi$ inverts one generator of ${\mathbb Z}^3$ and induces an automorphism of infinite order on the ${\mathbb Z}^2$ spanned by the other two generators. $\endgroup$
    – Derek Holt
    Jul 17, 2017 at 11:01
  • $\begingroup$ @ColinReid Do you happen to know of a nilpotent example? (namely a f.g torsion-free nilpotent group that has a finite noncentral conjugacy class) $\endgroup$
    – Itamar Vigdorovich
    Oct 18, 2020 at 18:48

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Such groups do not exist by Schur's theorem https://chiasme.wordpress.com/2015/01/07/a-theorem-of-schur-on-commutator-subgroup/

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  • $\begingroup$ Thank you for your answer, and sorry for the ambiguity in the question. The question I intended to ask (and the follow-up question) have been answered in the comments. $\endgroup$
    – Mark Grant
    Jul 17, 2017 at 16:11

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