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An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center.

I would like to know what is the motivation behind this definition.

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closed as off-topic by Andreas Blass, user44191, Sean Lawton, Chris Godsil, Pace Nielsen Mar 13 at 20:46

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    $\begingroup$ Since you are quoting – without saying it, might I add – word-for-word the nLab page anabelian group, I feel compelled to suggest clicking on the link to anabelian geometry and read it...... $\endgroup$ – Najib Idrissi Mar 12 at 16:31
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    $\begingroup$ This notion of "anabelian" seems to be exclusively used in algebraic geometry. In group theory "anabelian" appears in work of Nikolov and Segal, to denote a finite (or profinite) group with no abelian composition factor. These are distinct notions. $\endgroup$ – YCor Mar 12 at 20:47
  • $\begingroup$ This is not a motivation, but it is worth remarking that according to the definition given in the question, anabelian groups are those non-triviaal groups in which the conjugacy class of the identity is the only finite conjugacy class. A reasonably natural collection of groups to study, but from an algebraic point of view, not a very restrictive condition. $\endgroup$ – Geoff Robinson Mar 13 at 13:21
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I recommend you this article by Kirsten Wickelgren to supplement the links Najib provides, which establishes the motive for constructing anabelian objects.

In an Anabelian Scheme, the solutions are controlled not by the usual algebraic manipulations but rather by using the loops on the complex solutions together with a Galois group. This difference opens up new possibilities for understanding the solutions, which is one reason to care about Anabelian Schemes.

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This is not the context in which the word 'anabelian' arose, so it is not 'the' motivation, but there is at least 'a' motivation for an equivalent definition. Given an infinite group $G$, one can define the 'virtual centre' or 'FC-centre' to be the set of elements whose centralizer has finite index. The class of groups defined as 'anabelian' in the question are then the infinite groups with trivial virtual centre.

As an example of why you might want to restrict to this class of groups: if $G$ is an infinite group with trivial virtual centre, there is a well-behaved notion of the group $A$ of virtual automorphisms of $G$. The group $A$ is then universal for the class of groups $H$ that contain $G$ as a commensurated subgroup (that is, a subgroup such that $|G:G \cap hGh^{-1}|$ is finite for all $h \in H$) with trivial centralizer. See for instance:

Y. Barnea, M. Ershov and T. Weigel, Abstract commensurators of profinite groups, Trans. AMS 363 (2011), 5381-5417.

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