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Can one characterize the amenable groups that have the property that the centralizer of every non-identity element is cyclic?

For example, must they be solvable?

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    $\begingroup$ Which elements of the free group have noncyclic centralizers? $\endgroup$ – Goldstern Apr 26 '13 at 22:26
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    $\begingroup$ @Goldstern: Nonabelian free groups are not amenable. $\endgroup$ – Misha Apr 26 '13 at 23:22
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    $\begingroup$ @solovei: Now, this, is a reasonable question. For elementary amenable groups, the answer, I think, affirmative (i.e., all such groups are solvable). For nonelementary amenable groups, a comprehensive answer is well beyond the reach of the present technology. However, for the specific question about solvability, I would bet on a counterexample. For instance, I see nothing to prevent existence of a f.g. amenable group with trivial center, where every proper subgroup is cyclic (a version of Tarski monster). (If you ask A.Ol'shansky, he might be able to construct one for you.) $\endgroup$ – Misha Apr 26 '13 at 23:39
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    $\begingroup$ Sorry, I misread the question as "Can one characterize amenable groups BY the following property" (which would be silly). $\endgroup$ – Goldstern Apr 27 '13 at 0:24
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    $\begingroup$ Am I the only one who is bugged by questions that start in the title instead of being self-contained in the body? $\endgroup$ – Arturo Magidin Apr 27 '13 at 5:18

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