Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification (Q658526)

The author studies the action of the mapping class group on the integral homology of finite covers of a topological surface \(\Sigma\). The homological representation of the mapping class group carries a great deal of information about mapping classes but also forgets a lot of information. If one is willing to consider the action of mapping classes on the homology of finite covers of \(\Sigma\), one can recover much of the information as one looks at higher and higher covers of \(\Sigma\). The construction of a faithful infinite-dimensional representation of the mapping class group detects the Nielsen-Thurston classification of each mapping class. The author then discusses some examples that occur in the theory of braid groups and develops an analogous theory for automorphisms of free groups. The article ends with some open problems.