Algebraic and definable closure in free groups

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

Dependence and algebraicity over subgroups of free groups

“…In [OHV16] the term algebraic closure is used in the more traditional sense to denote the set of all group elements that depend on H, similar to what we call dep H according to Definition 2.1 (1), but with the restriction that the corresponding equations should have only finitely-many solutions. Note, however, that when g is algebraic over H according to [OHV16] then the fact that it satisfies an equation over H with finitely-many solutions does not imply that | H, g : H| < ∞, in contrast to the analogous simple algebraic extension in field theory, which is always a finite extension.…”