Algebraic and definable closure in free groups

Ould Houcine, Abderezak; Vallino, Daniele

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2525-2563.

Nous étudions la clôture algébrique et sa relation avec la clôture définissable dans les groupes libres et plus généralement dans les groupes hyperboliques sans torsion. Pour un groupe hyperbolique sans torsion $\Gamma$ et un sous-groupe non abélien $A$ de $\Gamma$, on décrit $\Gamma$ comme un groupe constructible à partir de la clôture algébrique de $A$ au-dessus de sous-groupes cycliques. On en déduit en particulier, que la clôture algébrique de $A$ est de type fini, quasiconvexe et hyperbolique.

Supposons que $\Gamma$ est libre. Alors la clôture définissable de $A$ est un facteur libre de la clôture algébrique de $A$ et les rangs de ces groupes est borné par celui de $\Gamma$. On montre que la clôture algébrique de $A$ coïncide avec le groupe sommet contenant $A$ dans la décomposition JSJ cyclique et malnormal de $\Gamma$ relative à $A$. Si le rang de $\Gamma$ est plus grand que $4$, on démontre que $\Gamma$ a un sous-groupe $A$ dont la clôture définissable est un sous-groupe propre de la clôture algébrique de $A$. Cela répond en particulier à une question de Sela.

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

Classification : 03C68, 11U09, 20E05, 20F67, 05E18
Keywords: Definable closure, algebraic closure, free groups, hyperbolic groups, JSJ-decompositions
Mot clés : Clôture définissable, clôture algébrique, groupes libres, groupes hyperboliques, JSJ-décompositions

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