Geometriae Dedicata

, Volume 156, Issue 1, pp 13–30 | Cite as

Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification

Original Paper

Abstract

We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.

Keywords

Mapping class groups Braid groups Nielsen–Thurston classification Representations of the mapping class group Automorphisms of free groups 

Mathematics Subject Classification (2010)

57M10 57M99 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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