## Finiteness properties for handlebody mapping class groups |

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### Contents

Local Finiteness of DAC | 81 |

Finiteness of the Quotient DACr | 86 |

Homological Stability | 97 |

Copyright | |

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### Common terms and phrases

2-sphere 3-manifold analogous arc of intersection arc representatives arc segments arc system automorphism boundary circles boundary components Chapter classes of arcs classes of diffeomorphisms connected construction contractible corresponding Daniel Quillen defined definition deformation retracts Dehn twists denote diffeomorphism disc representatives disc system discs Dj endpoints essential discs Euler characteristic fact Figure finite index finite number finitely presented fixes follows genus g given global bound group of isotopy handlebody mapping class Hence homological stability homotopy equivalent identity induction inessential isomorphism isotopy classes isotopy extension Lemma locally finite mapping class group maximal system pair of pants patched handlebody pointwise preserves properly embedded punctures quotient restriction result set of arcs set of discs set of representatives short exact sequence simple system simplex simplicial complex simplicial map simply-connected splitting St(v stably subcomplex subgroup subset surjective system of arcs system of discs Theorem vertex vertices Xg,i