We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
In chapter 1, we introduce the basic notions and well known facts related to find the group presentations for braid groups on surfaces. And we observe the relation between braid groups and mapping class groups on surfaces.
In chapter 2, we review the several representations on the classical braid group which is the braid group on disc. In particular, we focus on the homology linear representation because all known representations for the classical braid group can be regarded as the special cases of the homology linear representation. We also discuss more deeply about Burau representation and Lawrence-Krammer-Bigelow representations as special cases.
In chapter 3, we first try for the braid groups on surfaces to follow the way how the homology linear representations of the classical braid group was constructed and prove that this naive extension must lead us to the undesired result because the resultant is almost trivial. After that, we propose the new way to extend the homology linear representation to the braid groups on surfaces. Finally we prove that our proposed representation is actually an extension of the classical braid case.