  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 G and a nonabelian subgroup A of G, we describe G
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 G 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
G. We prove that the algebraic closure of A coincides with the vertex group
containing A in the generalized cyclic JSJ-decomposition of G relative to A. If
the rank of G is bigger than 4, then G 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.

Group Theory;Logic