Set theory is interpretable in the automorphism group of a free group

Abstract

In 1976 S. Shelah posed the following problem: for which variety V of algebras the automorphism group of any free algebra F from V of "large" infinite rank interprets by means of first-order logic set theory (according to his results, for every variety V the endomorphism semi-group of F interprets set theory if rank(F) is an infinite cardinal greater than the power of the language of V). There are examples of varieties for which the answer is negative; one such an example, the variety of all algebras in empty language, is due to Shelah (1973). The author earlier showed that the answer is positive for any variety of vector spaces over a fixed division ring. In the present paper it is proved that the same holds for the variety of all groups: the automorphism group of any infinitely generated free group F interprets set theory. It follows, in particular, that the group Aut(F) is as undecidable as possible.