We prove that (Z, +, 0) has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank ω. Additionally, our methods yield other superstable expansions such as (Z, +, 0) equipped with the set of factorial elements.2010 Mathematics Subject Classification. 03C45.

International audienceWe show that any nonabelian free group F is strongly @ 0-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under Aut.F/. We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not strongly @ 0-homogeneous. 1. Introduction Since the works of Sela [Sel1]–[Sel5] and Kharlampovich and Myasnikov [KM] on Tarski's problem, which showed that finitely generated free groups of rank at least 2 all have the same first-order theory, there has been renewed interest in model-theoretic questions about free groups. With Sela's work, it has become clear that techniques of geometric group theory provide extremely effective ways to tackle these questions. This can be seen in subsequent results such as [Sel5], where Sela showed that the theory of the free group is stable, or in [Per2], where the elementary subgroups of a free group of finite rank are shown to be exactly its nonabelian free factors. Moreover, the geometric nature of the tools often allows these results to be generalized to the class of torsion-free hyperbolic groups (see [Sel4]). In this paper, we apply these techniques to study types of elements and homogeneity in torsion-free hyperbolic groups, particularly in free groups and hyperbolic surface groups. The type of a tuple N a in a structure is the set of all first-order formulas it realizes. More formally, let L be a first-order language, let M be an L-structure, and let B be a subset of M. (For basic definitions of model theory, the reader is referred to [CK], [Mar], or the short survey given in [Cha].) If N a is a k-tuple of elements of M, the type tp M. N a=B/ of N a over B in M is the set of all formulas with k free variables and parameters in B which N a realizes in M. An L-structure M is strongly @ 0-homogeneous if, for any pair of finite tuples N a; N a 0 with tp M. N a=;/ D tp M. N a 0 =;/, there is an automorphism of M sending N a to N a 0

We give a complete characterization of the forking independence relation over any set of parameters in the free groups of finite rank, in terms of the JSJ decompositions relative to those parameters.