On kernel of quantum representations of mapping class groups

This talk will be centered around some finite dimensional complex representations of mapping class groups of surfaces. More precisely, we will discuss quantum representations of mapping class groups arising from Witten-Reshetikhin–Turaev Topological Quantum Fields Theories. These are projective unitary finite dimensional complex representations of mapping class groups indexed by an integer called level. When the level is a prime number, the image of the representation lands in the integral points of an algebraic group G and is Zariski dense in G. One important open problem is to know if this image has finite index in G(Z). As I will explain, for a fixed prime level p, knowing the kernel of the representation might help knowing if the image is arithmetic or not. I will explain a joint work with Renaud Detcherry where we can give some information about this kernel, more precisely we compute the two first terms of the so-called h-adic approximation of the representation (which is a sequence of finite groups approximation of the representation).