Maximal hyperbolic towers and weight in the theory of free groups

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: We construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.