Hall’s Theorem for Limit Groups

By Henry Wilton. 2006

85 citations recorded by the source corpus.

Abstract.A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cf57b31882513e6120967d9682714f1091572049

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the finite presentation of subdirect products and the nature of residually free groups (2012, background): “The projection Hi of H to Λi is finitely generated and Wilton’s theorem [40] tells us that Λi is subgroup separable, so we can determine algorithmically if gi ∈Hi.”
Determining Fuchsian groups by their finite quotients (2014, background): “Wilton [55] proved that if Γ is a limit group and H is a finitely-generated subgroup of Γ, then there is a subgroup Γ0 < Γ of finite index that contains H and retracts onto it.”
RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY (2009, background): “Combined with subgroup separability results for virtually special groups that are hyperbolic relative to abelian subgroups we are able to recover Wilton’s result that limit groups are subgroup separable [46].”
Subgroups of direct products of limit groups (2007, background): “…should consult the introductions in [1, 6], the original papers of Sela [22, 23, 24], or those of Kharlampovich and Myasnikov [18, 19, 20] where the subject is approached from a perspective more in keeping with traditional combinatorial group theory; a further perspective is developed in [16].”