Derived subgroup not is purely definable

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., commutator subgroup) does not always satisfy a particular subgroup property (i.e., purely definable subgroup)
View subgroup property satisfactions for subgroup-defining functions

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View subgroup property dissatisfactions for subgroup-defining functions

History

This is based on as yet unpublished result of Bestvina and Feighn (referred to here).

Statement

The derived subgroup of a group need not be a purely definable subgroup.

More specifically, the commutator subgroup of the free group of rank two is not a purely definable subgroup.

Related facts

• Center is purely definable
• Finite upper central series members are purely definable