Peripheral fillings of relatively hyperbolic groups

Abstract

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π 1 (M ). The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G 'almost' have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Various applications of these results are discussed.

Dehn fillings and elementary splittings

“…Agol and Lackenby's 6-Theorem [Ago00,Lac00] shows that the group-theoretic conclusions can be obtained by a softer, more combinatorial argument. This work was part of the inspiration for results about purely group-theoretic Dehn filling obtained by Osin [Osi07] and the authors [GM08], and generalized still further in the work of Dahmani-Guirardel-Osin [DGO11]. These results all have a "hyperbolic-like" conclusion analogous to that of the 6-Theorem.…”

Fillings, finite generation and direct limits of relatively hyperbolic groups

“…In [16], Osin proved the following theorem: Theorem 6. [16, Theorem 1.1] Suppose that G is hyperbolic relative to the system of subgroups {H λ } λ∈Λ .…”

“…In Section 3 we explain how the main result in Osin's paper [16] about Dehn filling in relatively hyperbolic groups follows from the version where the relatively hyperbolic groups are assumed to be finitely generated. (In [10] the authors proved this finitely generated version under the additional assumption that the group is torsion-free.)…”

Peripheral fillings of relatively hyperbolic groups

“…The other two assertions of the theorem are proved in a similar way. All details can be found in [27].…”