FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

Philipp Lücke,Saharon Shelah

doi:10.1017/fms.2014.9

Abstract

AbstractGiven a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

Highlights

Given an infinite cardinal λ, we say that a group G is the automorphism group of a λ-structure if there is a first-order language L and an L-structure M such that the cardinality of the signature of L and the cardinality of the domain of M are at most λ and the group Aut(M) consisting of all automorphisms of c The Author(s) 2014
Let λ be a singular cardinal of uncountable cofinality such that there is no free group of rank greater than λ that is the automorphism group of a λ-structure
Realizing limits of groups as automorphism groups we show that certain limit objects in the category of groups can be realized as the automorphism group of a structure whose cardinality only depends on the size of the objects appearing in the corresponding limit construction

Summary

Introduction

Given an infinite cardinal λ, we say that a group G is the automorphism group of a λ-structure if there is a first-order language L and an L-structure M such that the cardinality of the signature of L and the cardinality of the domain of M are at most λ and the group Aut(M) consisting of all automorphisms of c The Author(s) 2014. The methods developed in the proof of Theorem 1.4 allow us to show that the cardinal arithmetic assumption that λ = λא0 is consistently not necessary for the existence of a free group of rank 2λ that is the automorphism group of a λ-structure. The above results raise the question whether the existence of a cardinal λ of uncountable cofinality with the property that no free group of rank greater than λ is the automorphism group of a λ-structure is even consistent with the axioms of ZFC.

Results

Conclusion