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This is based on as yet unpublished result of Bestvina and Feighn (referred to here).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