We use hyperbolic towers to answer some model-theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_{0}$ but that there is a finitely generated model which omits $p^{}_{0}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is (...) not necessarily homogeneous. (shrink)

In this short note we prove that a definable set X over is superstable only if.

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.

We prove that any torsion-free, residually finite relatively free group of infinite rank is not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} -homogeneous. This generalizes Sklinos’ result that a free group of infinite rank is not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} -homogeneous, and, in particular, gives a new simple proof of that result.