### Abstract

For a Haken 3-manifold M with incompressible boundary, we prove that the mapping class group H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in H(M) exist is given. All results are given for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in ∂M such that ∂M - F is incompressible, then the classifying space BDiff(M rel F) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.

Original language | English |
---|---|

Pages (from-to) | 275-301 |

Number of pages | 27 |

Journal | Pacific Journal of Mathematics |

Volume | 188 |

Issue number | 2 |

Publication status | Published - 1999 Apr 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*188*(2), 275-301.

**Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds.** / Hong, Sungbok; McCullough, Darryl.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 188, no. 2, pp. 275-301.

}

TY - JOUR

T1 - Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds

AU - Hong, Sungbok

AU - McCullough, Darryl

PY - 1999/4/1

Y1 - 1999/4/1

N2 - For a Haken 3-manifold M with incompressible boundary, we prove that the mapping class group H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in H(M) exist is given. All results are given for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in ∂M such that ∂M - F is incompressible, then the classifying space BDiff(M rel F) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.

AB - For a Haken 3-manifold M with incompressible boundary, we prove that the mapping class group H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in H(M) exist is given. All results are given for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in ∂M such that ∂M - F is incompressible, then the classifying space BDiff(M rel F) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.

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M3 - Article

VL - 188

SP - 275

EP - 301

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -