Quid of Mapping Class Groups & their Representations
The mapping class group Mod(Σ) of a surface Σ is a fundamental object in low-dimensional topology, as it acts on many interesting geometric spaces: the space of (marked) hyperbolic metrics, the SL₂(ℂ)-character variety, the space of measured foliations, etc. However, as interesting as mapping class groups are, many of their group-theoretic aspects are still poorly understood. For example: the question of whether or not Mod(Σ) admits an injective linear representation remains wide open.
In this talk, we will discuss the theory of generators and relations for surface mapping class groups and highlight some recent results by Korkmaz on its low-dimensional representations. If time allows, we will illustrate the techniques used by Korkmaz by establishing a lower bound for the dimension of a nontrivial unitary representation of PMod(Σ).