TY - JOUR

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

AU - Hong, Sungbok

AU - McCullough, Darryl

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1999/4

Y1 - 1999/4

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.

UR - http://www.scopus.com/inward/record.url?scp=0039528874&partnerID=8YFLogxK

U2 - 10.2140/pjm.1999.188.275

DO - 10.2140/pjm.1999.188.275

M3 - Article

AN - SCOPUS:0039528874

SN - 0030-8730

VL - 188

SP - 275

EP - 301

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -