Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group

DOI10.1112/S0024610700008760zbMATH Open0969.03049arXivmath/9712249MaRDI QIDQ4520455

Vladimir Tolstykh

Publication date: 13 December 2000

Published in: Journal of the London Mathematical Society (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/math/9712249

zbMATH Keywords

first-order theoryfree groupautomorphism groupselementary equivalenceelementary theorysecond-order theorymutual interpretability

Mathematics Subject Classification ID

Model-theoretic algebra (03C60) Automorphism groups of groups (20F28) Free nonabelian groups (20E05) Models of arithmetic and set theory (03C62) Second- and higher-order model theory (03C85)

Recommendations

• Unnamed Item 👍 👎
• Unnamed Item 👍 👎
• FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES 👍 👎
• The automorphism groups of relatively free groups of infinite rank 👍 👎
• On genericity and weight in the free group 👍 👎
• The automorphisms of endomorphism semigroups of relatively free groups 👍 👎
• On the logical strength of the automorphism groups of free nilpotent groups 👍 👎
• Automorphisms of the endomorphism semigroup of a free commutative dimonoid 👍 👎
• AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA 👍 👎
• On automorphisms of relatively free groups 👍 👎

This page was built for publication: Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group