Submission date: 30 August 2011.

Abstract:

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.

Mathematics Subject Classification: 20B07, 20E05, 12L12, 20F65, 20E08, 20F67, 20F70

Keywords and phrases: free groups, torsion-free hyperbolic groups, algebraic closure, definable closure

Full text arXiv 1108.5641: pdf, ps.