Peripheral fillings of relatively hyperbolic groups

By Denis V. Osin. 2005

147 citations recorded by the source corpus.

In this paper 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 π1(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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1a40cddf19ebd8c99396ca821d5018e49cb36613

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The virtual Haken conjecture (2012, result): “If one quotes the filling theorem from [37] (which holds in the presence of torsion) instead of [17], then the proof from [1] works as written.”
Acylindrically hyperbolic groups (2013, background): “For details and relation to 3-dimensional topology and Thurston’s work we refer to [40].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, methodology): “2 in Section 3 of [105], almost verbatim after few changes in notation and terminology.”
Dehn filling in relatively hyperbolic groups (2006, background): “Second, and more seriously, Osin makes no assumption of torsion-freeness in [ 36 ].”
Residual finiteness, QCERF, and fillings of hyperbolic groups (2008, result): “We collect here some results about filling from [8] (see also Osin [19]):”
RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY (2009, background): “This work joins a variety of generalizations of small-cancellation theory including works of [38, 33, 39, 16].”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, background): “In the aforementioned paper, the authors introduced the notion of hyperbolically embedded subgroup of a group and extended the relatively hyperbolic Dehn filling theorem [18, 27], thereby constructing many interesting quotients of groups such as mapping class groups.”
GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, background): “Denis Osin (2006).”
Normal automorphisms of relatively hyperbolic groups (2008, methodology): “Indeed we use the group-theoretic version of Dehn surgery introduced in [18, 19, 32] and ‘component analysis’ developed in [33, 27], while Metaftsis and Sykiotis employed the Bestvina-Paulin approach [5, 34] based on ultralimits and group actions on R-trees.”
Stable commutator length in word-hyperbolic groups (2006, unknown): “[23], [31]).”
Relative hyperbolicity and relative quasiconvexity for countable groups (2008, result): “3.15]. We remark that if S is a relative generating set rather than a generating set, the result still holds and can be derived as a corollary of [ Osi07 , Prop.”