Algebraic & 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 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

Model theory of finite and pseudofinite groups

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory

The monomorphism problem in free groups

Abstract Let F be a free group of finite rank. We say that the monomorphism problem in F is decidable if there is an algorithm such that, for any two elements u and v in F , it determines whether there exists a monomorphism of F that sends u to v. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem. 2000 Mathematics Subject Classification: 20E05, 68Q25