|T|+-resplendent models and the Lascar group

Lascar in [4] introduced the group Autf(N/A) of strong automorphisms over A, a normal subgroup of the group Aut(N/A) of all automorphisms overA of a modelN containingA. The quotient Aut(N/A)/Autf(N/A) is independent of the choice of N (for a big saturated model N and a small subset A ⊆ N ) and it is now called the Lascar group over A. Lascar showed in [4] that in the case of a very large class of theories, called by him G-compact, the group carries a compact Hausdorff topology. Recently the Lascar group has received a lot of attention, particularly because of its importance for simple theories and hyperimaginaries and because of the discovery of non-G-compact theories. It is a compact (not necessarily Hausdorff) topological group for any firstorder theory, even in a non G-compact one. In the case of the theory of an algebraically closed field it corresponds to the absolute Galois group over the field generated by A, and it is a profinite group. We refer to [5, 3, 8] for more details. The presentation of the group in [4] is done in the framework of an uncountable saturated model N of an arbitrary countable complete first-order theory and a finite subset A ⊆ N . It is straightforward to generalize it to any complete first-order theory T and any saturated model N of T with |N | > |T |. Thus it always can be constructed working in the monster model C of T . Although the details have not been written, it is generally acknowledged that instead of saturated models one can use special models of the right cardinality. For instance, Ziegler observes in [8] that a special model N such that cf(|N |) > 2 | is sufficient. The inconvenience of working with saturated models is that for some theories its existence can not be proven without extra set theoretical hypotheses. On the other hand special models do always exist. We have noticed that there is a more general class of models where the Lascar group naturally arises: the class of |T |-resplendent models. Moreover the properties of the group of strong automorphisms can be understood more easily working with these models. The notion of resplendency has been introduced by Barwise and Schlipf in [1]. Poizat in [6] defined and studied the more general notion of κ-resplendency. In Section 2 we summarize the main facts. |T |-resplendent models generalize (in the right cardinality) saturated and special models, and in the case of stable theories they coincide with saturated models. In unstable theories there are many |T |-resplendent models which are neither saturated nor special.