Peripheral fillings of relatively hyperbolic groups.

*(English)*Zbl 1116.20031The author first notes that Thurston’s Dehn filling theorem for 3-manifolds implies the following group theoretic result: If \(G\) is the fundamental group of a complete finite volume hyperbolic 3-manifold and if \(H_1,\dots,H_k\) are the cusp subgroups of \(G\), then there exists a finite subset \(\mathcal F\) of \(G\) such that for any collection of primitive elements \(x_i\) in \(H_i\setminus\mathcal F\), the quotient group \(G/\langle x_1,\dots,x_k\rangle^G\) is word hyperbolic. (Here, if \(S\) is a subset of \(G\), \(\langle S\rangle^G\) denotes the normal closure of \(S\) in \(G\).)

In the paper under review, the author gives the following generalized setting of this result. Instead of the fundamental group of a complete finite volume hyperbolic manifold, he considers a general relatively hyperbolic group \(G\). In this setting, the algebraic analogue of Dehn filling is defined as follows. Let \(\{H_\lambda\}_{\lambda\in\Lambda}\) be a collection of subgroups of \(G\). To each collection of groups \(\mathcal N=\{N_\lambda\}_{\lambda\in\Lambda}\), where for each \(\lambda\), \(N_\lambda\) is a normal subgroup of \(H_\lambda\), one associates the quotient group \[ G(\mathcal N)=G/\langle\bigcup_{\lambda\in\Lambda}N_\lambda\rangle^G. \] Furthermore, instead of taking a single element \(x_i\) in \(H_i\), the author considers normal subgroups generated by arbitrary subsets of the cusp subgroups.

The main result of this paper is then the following Theorem: Let \(G\) be a group which is relatively hyperbolic with respect to a collection of subgroups \(\{H_\lambda\}_{\lambda\in\Lambda}\). Then there exists a finite subset \(\mathcal F\) of non-trivial elements of \(G\) with the following property. Let \(\mathcal N=\{N_\lambda\}_{\lambda\in\Lambda}\) be a collection of subgroups, where each \(N_\lambda\) is normal in \(H_\lambda\) and such that \(N_\lambda\cap\mathcal F=\emptyset\) for all \(\lambda\) in \(\Lambda\). Then, for each \(\lambda\) in \(\Lambda\), the natural map \(H_\lambda/N_\lambda\to G(\mathcal N)\) is injective, and the quotient group \(G(\mathcal N)\) is hyperbolic relative to the collection \(\{H_\lambda/N_\lambda\}_{\lambda\in\Lambda}\). Moreover, for any finite subset \(S\subset G\), there exists a finite subset \(\mathcal F(S)\) of non-trivial elements of \(G\) such that the restriction of the natural homomorphism \(G\to G(\mathcal N)\) to \(S\) is injective whenever \(N_\lambda\cap\mathcal F(S)=\emptyset\) for all \(\lambda\in\Lambda\).

Note that in this theorem, \(G\) is not assumed to be finitely generated.

The proof of the theorem is combinatorial and uses techniques related to van Kampen diagrams.

The author deduces several interesting corollaries, among them the following Corollary: Suppose, in addition to the assumptions of the theorem, that \(G\) is finitely generated and \(H_\lambda/N_\lambda\) is hyperbolic for each \(\lambda\in\Lambda\). Then, \(G(\mathcal N)\) is hyperbolic.

The author also obtains algebraic applications of the theorem. For instance, he proves that any almost malnormal quasi-convex subgroup of a hyperbolic group “almost has the congruence extension property”. This is related to some work of Brian Bowditch. He also obtains non-trivial examples of fully residually hyperbolic groups, and he proves that the following two assertions are equivalent: (1) All hyperbolic groups are residually finite. (2) If a finitely generated group \(G\) is hyperbolic relative to a collection of residually finite subgroups, then \(G\) is residually finite.

Note that the question of whether all hyperbolic groups are residually finite is an open question.

In the paper under review, the author gives the following generalized setting of this result. Instead of the fundamental group of a complete finite volume hyperbolic manifold, he considers a general relatively hyperbolic group \(G\). In this setting, the algebraic analogue of Dehn filling is defined as follows. Let \(\{H_\lambda\}_{\lambda\in\Lambda}\) be a collection of subgroups of \(G\). To each collection of groups \(\mathcal N=\{N_\lambda\}_{\lambda\in\Lambda}\), where for each \(\lambda\), \(N_\lambda\) is a normal subgroup of \(H_\lambda\), one associates the quotient group \[ G(\mathcal N)=G/\langle\bigcup_{\lambda\in\Lambda}N_\lambda\rangle^G. \] Furthermore, instead of taking a single element \(x_i\) in \(H_i\), the author considers normal subgroups generated by arbitrary subsets of the cusp subgroups.

The main result of this paper is then the following Theorem: Let \(G\) be a group which is relatively hyperbolic with respect to a collection of subgroups \(\{H_\lambda\}_{\lambda\in\Lambda}\). Then there exists a finite subset \(\mathcal F\) of non-trivial elements of \(G\) with the following property. Let \(\mathcal N=\{N_\lambda\}_{\lambda\in\Lambda}\) be a collection of subgroups, where each \(N_\lambda\) is normal in \(H_\lambda\) and such that \(N_\lambda\cap\mathcal F=\emptyset\) for all \(\lambda\) in \(\Lambda\). Then, for each \(\lambda\) in \(\Lambda\), the natural map \(H_\lambda/N_\lambda\to G(\mathcal N)\) is injective, and the quotient group \(G(\mathcal N)\) is hyperbolic relative to the collection \(\{H_\lambda/N_\lambda\}_{\lambda\in\Lambda}\). Moreover, for any finite subset \(S\subset G\), there exists a finite subset \(\mathcal F(S)\) of non-trivial elements of \(G\) such that the restriction of the natural homomorphism \(G\to G(\mathcal N)\) to \(S\) is injective whenever \(N_\lambda\cap\mathcal F(S)=\emptyset\) for all \(\lambda\in\Lambda\).

Note that in this theorem, \(G\) is not assumed to be finitely generated.

The proof of the theorem is combinatorial and uses techniques related to van Kampen diagrams.

The author deduces several interesting corollaries, among them the following Corollary: Suppose, in addition to the assumptions of the theorem, that \(G\) is finitely generated and \(H_\lambda/N_\lambda\) is hyperbolic for each \(\lambda\in\Lambda\). Then, \(G(\mathcal N)\) is hyperbolic.

The author also obtains algebraic applications of the theorem. For instance, he proves that any almost malnormal quasi-convex subgroup of a hyperbolic group “almost has the congruence extension property”. This is related to some work of Brian Bowditch. He also obtains non-trivial examples of fully residually hyperbolic groups, and he proves that the following two assertions are equivalent: (1) All hyperbolic groups are residually finite. (2) If a finitely generated group \(G\) is hyperbolic relative to a collection of residually finite subgroups, then \(G\) is residually finite.

Note that the question of whether all hyperbolic groups are residually finite is an open question.

Reviewer: Athanase Papadopoulos (Strasbourg)

##### MSC:

20F65 | Geometric group theory |

20F67 | Hyperbolic groups and nonpositively curved groups |

20F06 | Cancellation theory of groups; application of van Kampen diagrams |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

20E26 | Residual properties and generalizations; residually finite groups |

##### Keywords:

Dehn fillings; peripheral fillings; quasi-convex subgroups; cusp subgroups; relatively hyperbolic groups; word hyperbolic groups; van Kampen diagrams; residually hyperbolic groups; finitely generated groups##### References:

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