We give two slight generalizations of results of Poizat about elementary theories of groups obtained by free constructions. The first-one concerns generic types and the non-superstability of such groups in many cases. The second-one concerns the connectedness of most free products of groups without amalgamation.

A first order theory T of power λ is called unidimensional if any twoλ+-saturated models of T of the same cardinality are isomorphic. We prove here that such theories are superstable, solving a problem of Shelah. The proof involves an existence theorem and a definability theorem for definable groups in stable theories, and an analysis of their relation to regular types.

A first order theory T of power λ is called unidimensional if any twoλ+-saturated models of T of the same cardinality are isomorphic. We prove here that such theories are superstable, solving a problem of Shelah. The proof involves an existence theorem and a definability theorem for definable groups in stable theories, and an analysis of their relation to regular types.

We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.