ALGEBRAIC AND DEFINABLE CLOSURE IN FREE GROUPS

We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group Gamma and a nonabelian subgroup A of Gamma, we describe Gamma as a constructible group from the algebraic closure of A along cyclic subgroups. In particular, it follows that the algebraic closure of A is finitely generated, quasiconvex and hyperbolic. Suppose that Gamma is free. Then the definable closure of A is a free factor of the algebraic closure of A and the rank of these groups is bounded by that of Gamma. We prove that the algebraic closure of A coincides with the vertex group containing A in the generalized malnormal cyclic JSJ-decomposition of Gamma relative to A. If the rank of Gamma is bigger than 4, then Gamma has a subgroup A such that the definable closure of A is a proper subgroup of the algebraic closure of A. This answers a question of Sela.