2011/08/29 by Abderezak Ould Houcine, Houcine, A. Ould, Daniele Vallino +1 · 1 citation
Computer Science · Mathematics · #12L12 #20B07 #20E05 #20E08 #20F65 #20F67 #20F70 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1108.5641
openalex publication_date 2011/08/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.