Classifying totally categorical groups (and others) (Q1919528)

The author studies the structure of groups definable in a unidimensional, one-based theory \(T\) in which every minimal type is locally finite; any totally categorical theory satisfies these conditions. A connected group \(\wedge\)-definable in \(T\) is said to be basic; by Hrushovski-Pillay's theorem, every basic group is abelian. A basic group is said to be irreducible if its generic is; here an element of a group is called irreducible if there are no \(b,c \in \text{acl} (a)\) of smaller rank with \(a \in \text{acl} (bc)\). Main result: for every irreducible basic \(H\), there exists an irreducible basic supergroup \(G\) of \(H\) in \(\text{acl}^{\text{eq}} (H)\) -- the so-called modular closure of \(H\) -- such that, for any \(n < \omega\), every connected subgroup of \(G^n\) is the connected component of a subgroup linearly defined over the ring of quasi-endomorphisms of \(G\). Thus \(H\) is a subgroup of a \(\wedge\)-definable essentially pure module, which is moreover ``almost'' quantifier eliminable. The results extend the analysis of \textit{S. Buechler} from his preprint [Vaught's conjecture for superstable theories of finite rank (1992)] to the case of a unidimensional, one-based theory with all minimal types locally finite. The key point is the fact that there is a bound for the ranks of irreducibles algebraic in the group. Buechler used the few models assumption for that; the author uses instead the local finiteness assumption -- the modular closure is found as a group with irreducible generics of maximal rank. She gives an example which shows that the local finiteness assumption cannot be omitted. For a totally categorical \(H\) of \(U\)-rank 2, an explicit construction for the modular closure of \(H\) is given.