Two remarks on elementary theories of groups obtained by free constructions

We give two slight generalizations of results of Poizat about elementary theories of groups obtained by free constructions. The firstone 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. First-order theories of free products of groups have been recently investigated in [JS10] and [Sel10], with some transfers of arguments from free groups to free products of groups. We expect that some of this work transfers further to more general classes of groups obtained by free constructions. In this modest and short note we will make slight generalizations of early arguments of Poizat on first-order theories of free groups. By a group obtained by a free construction, we mean a free product of groups with amalgamation, or an HNN -extension. Such groups are best analized by the Bass-Serre theory of actions on trees, and we refer to [OH08] for links with Model Theory. Here our approach is purely algebraic and will adapt (and explain) some arguments contained in [Poi83]. We recall that one of the main accomplishements of [Sel10] is a proof of the stability of the free product of two stable groups. Conversely, we point out the following question as a possibly difficult one. Question 1. Can one have a free product of groups G ∗ H with a stable theory, but with the factor G unstable? We expect that the stability result in [Sel10] generalizes to certain free products with amalgamation and certain HNN -extensions, and Question 1 also makes sense in these more general cases. When considering a stable group obtained by a free construction below, we will not assume the stability of the factors. In a first series of results we make a basic analysis of generic types in many groups obtained by free constructions, and derive the non-superstability of these groups in many cases. Non-abelian free groups were first shown to be not superstable by Gibone in his doctoral dissertation under the supervision