FREE GROUPS and AUTOMORPHISM GROUPS of INFINITE STRUCTURES

Abstract

Given a cardinal with, we show that there is a field of cardinality whose automorphism group is a free group of rank. 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 is the automorphism group of a field of cardinality whenever is a cardinal with. Moreover, we apply these techniques to show that consistently the assumption that is not necessary for the existence of a field of cardinality whose automorphism group is a free group of rank. Finally, we use them to prove that the existence of a cardinal of uncountable cofinality with the property that there is no field of cardinality whose automorphism group is a free group of rank greater than implies the existence of large cardinals in certain inner models of set theory.