Dehn filling in relatively hyperbolic groups.

*(English)*Zbl 1211.20038A finitely generated group is said to be word hyperbolic if it acts properly and co-compactly on a metric space in which every geodesic triangle is \(\delta\)-thin for some \(\delta\). Roughly speaking, a group is said to be hyperbolic relative to some collection \(\mathcal P\) of subgroups if the geometry of \(G\) is hyperbolic except for that part corresponding to \(\mathcal P\).

The authors accomplish three goals in this paper. They introduce a new space (the “cusped space”) for studying relatively hyperbolic groups; they construct a pair of useful bicombings on that space; and they extend Thurston’s Hyperbolic Dehn Surgery Theorem to the context of torsion-free relatively hyperbolic groups.

The cusped space is constructed by gluing a collection of “combinatorial horoballs” (one for each coset of each group in \(\mathcal P\)) onto the Cayley graph of \(G\). The resulting space is then hyperbolic precisely when \(G\) is hyperbolic relative to \(\mathcal P\). The space is a graph with the edge-length metric that allows the authors to adapt a number of constructions and results in word hyperbolic groups to the relative case.

Next the authors define preferred paths for the cusped space, which give rise to a bicombing that is a relatively hyperbolic version of a construction of I. Mineyev [Geom. Funct. Anal. 11, No. 4, 807-839 (2001; Zbl 1013.20034)]. The authors expect that since Mineyev’s work has many applications, their construction should allow many results to be extended to the relatively hyperbolic setting.

Finally, the authors apply the notion of preferred paths to prove a group theoretic analog of Dehn filling. A corollary of this theorem along with another result from this paper allow the authors to unify a number of known results.

The authors accomplish three goals in this paper. They introduce a new space (the “cusped space”) for studying relatively hyperbolic groups; they construct a pair of useful bicombings on that space; and they extend Thurston’s Hyperbolic Dehn Surgery Theorem to the context of torsion-free relatively hyperbolic groups.

The cusped space is constructed by gluing a collection of “combinatorial horoballs” (one for each coset of each group in \(\mathcal P\)) onto the Cayley graph of \(G\). The resulting space is then hyperbolic precisely when \(G\) is hyperbolic relative to \(\mathcal P\). The space is a graph with the edge-length metric that allows the authors to adapt a number of constructions and results in word hyperbolic groups to the relative case.

Next the authors define preferred paths for the cusped space, which give rise to a bicombing that is a relatively hyperbolic version of a construction of I. Mineyev [Geom. Funct. Anal. 11, No. 4, 807-839 (2001; Zbl 1013.20034)]. The authors expect that since Mineyev’s work has many applications, their construction should allow many results to be extended to the relatively hyperbolic setting.

Finally, the authors apply the notion of preferred paths to prove a group theoretic analog of Dehn filling. A corollary of this theorem along with another result from this paper allow the authors to unify a number of known results.

Reviewer: Gregory C. Bell (Greensboro)

##### MSC:

20F65 | Geometric group theory |

57M50 | General geometric structures on low-dimensional manifolds |

20F67 | Hyperbolic groups and nonpositively curved groups |

57M07 | Topological methods in group theory |

##### Keywords:

relatively hyperbolic groups; Dehn filling; finitely generated groups; Cayley graphs; word hyperbolic groups; cusped spaces; bicombings##### References:

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