ar X iv : 0 80 4 . 28 41 v 1 [ m at h . G R ] 1 7 A pr 2 00 8 Note on the Cantor - Bendixon rank of limit groups

We show that the Cantor-Bendixon rank of limit groups is finite as well as that of limit groups of linear groups. Let T be an universal theory. In what follows this will be just the universal theory of some linear group. 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 [2]. We reformulate the definition of Rk in the context of topological spaces. We recall the definition of the Cantor-Bendixson rank. 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. 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 n-QF-type of T . Then S 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. Fact 0.2 S n (T ) is a compact totally disconnected space.