Automorphisms of models of PA: the neglected casesThe Graduate Center, City University of New York, 365 Fifth Avenue, 10016 New York, USA firstname.lastname@example.org
Automorphism groups of countable recursively saturated models of PA were studied intensively in the 1990's. In a series of papers (joint and separate) by Lascar, Kaye, Kotlarski, Schmerl and the author, the important notion of arithmetic saturation was isolated and a series of results characterizing arithmetic saturation in group theoretic terms were proved. The notions involved include: generic automorphisms, maximal automorphisms, the small index property, maximal open subgroups, and the cofinality of the group. Most of this material is presented in . Much less has been done concerning models which are recursively saturated and not arithmetically saturated (we do not even have a good name for this class of models), and even less concerning short recursively saturated models. I will give a survey of this entire area of research and I will include some more recent results concerning the neglected cases, in particular the problem of extendabability of automorphisms to cofinal extensions. This last topic is joint work with Henryk Kotlarski.
Roman Kossak, James Schmerl, The Structure of Models of Peano Arithmetic, Oxford Logic Guides 50, Oxford University Press, 2006.