Note on the Cantor-Bendixson rank of limit groups

We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group. Let T be a universal theory. If T has at most countably many finitely generated models, then one can associate to each finitely generated model M of T an ordinal rank, denoted Rk(M), as defined in [7]. We reformulate the definition of Rk in the context of topological spaces. Let X be a compact Hausdorff topological space. Let D(X) = {x ∈ X |x is not isolated in X}. We define inductively D(X) on ordinals as follows: • D(X) = X , • D(X) = D(D(X)), • D(X) = ⋂ β<αD (X), for α a limit ordinal. The following fact can be easily extracted from [5]. Fact 0.1 There exists a least ordinal α such that D(X) = D(X) for any β > α. If X is separable then D(X) = ∅ or |D(X)| = 20 . Thus if X is countable then D(X) = ∅. In that case we define the CantorBendixson rank of x ∈ X , denoted CB(x), by CB(x) = α if and only if x ∈ D(X) \D(X). Let us return to our universal theory T . A complete n-QF-type (with respect to T ), where the length |x̄| = n, is a set p(x̄) of quantifier-free formulas such that for any quantifier-free formula ψ(x̄) either ψ ∈ p or ¬ψ ∈ p, and such that there is a model M of T having a tuple ā such that M |= p(ā). Let S n (T ) be the set of all completes n-QF-types of T . Then S qf n (T ) is equipped with a topology as follows. Take for basis open sets the sets of the form [ψ] = {p ∈ S n (T )|ψ ∈ p}, where ψ is a QF-formula. The following fact is a consequence of the compactness theorem and the proof proceeds in a similar way to that of [9, Théorème 4.04]. Fact 0.2 S n (T ) is a compact totally disconnected space.