Let ℱ be the class of complete, finitely axiomatizable ω-categorical theories. It is not known whether there are simple theories in ℱ. We prove three results of the form: if T∈ ℱ has a sufficently well-behaved definable set J, then T is not simple. All of our arguments assume that the definable set J satisfies the Mazoyer hypothesis, which controls how an element of J can be algebraic over a subset of the model. For every known example in ℱ, there (...) is a definable set satisfying the Mazoyer hypothesis. (shrink)

Let be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of (...) view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number. Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures. (shrink)

We investigate the structure of C-minimal valued groups that are not abelian-by-finite. We prove among other things that they are nilpotent-by-finite.