Vaught's conjecture for superstable theories of nite rank

In this paper we prove Vaught's conjecture for superstable theories in which each complete type has nite U? rank. The general idea is to associate with the theory an V ? deenable group G (called the structure group) which controls the isomorphism types of the models. We use module theory to show that when the theory has few countable models and M is a countable model there is a nice decomposition of G(M) as the sum of copies of nitely many nitely generated subgroups. This is the principle ingredient in the proof of the following main theorem, from which Vaught's conjecture follows immediately. Theorem 0.1 (Structure Theorem) Suppose that T is superstable of nite rank, has < 2 @ 0 many countable models, and M j = T is countable. Then there is a nite A M and a set J independent over A such that M is prime over A J and f stp(a=A) : a 2 J g is nite. 1 Preliminaries In Vau61] Vaught conjectured that a countable rst order theory has countably many or 2 @ 0 many countable models. In this paper we prove Vaught's conjecture for superstable theories in which each complete type has nite U? rank. The less common terminology is summarized in the next few paragraphs. Unless stated otherwise, all theories are assumed to be superstable. Such a theory is said to be of nite rank if every complete type has nite U-rank (U(?) denotes Lascar's rank, described in Mak84]). A theory is unidimensional if all nonalgebraic types are nonorthogonal (see, e.g., She90] or Bal88]). Since we always work in T eq