The non-realizability of braid groups by diffeomorphisms

Nick Salter (Columbia)

Abstract: Braid and mapping class groups are defined as *quotients* of groups of diffeomorphisms, and a line of questioning dating back to Nielsen asks when it is possible to pick “canonical” representatives for a group of mapping classes and so realize it as a group of diffeomorphisms. Kerckhoff showed that any finite group of mapping classes can be realized by diffeomorphisms, while Morita showed that no finite-index subgroup of the mapping class group of a closed surface of sufficiently large genus g can be realized. Morita’s method fails in the case of braid groups (mapping class groups of punctured disks) because the cohomology classes at the heart of the argument now vanish; Nariman later showed in a precise sense that there is no possible cohomological obstruction. In this talk I will explain work, joint with Bena Tshishiku, that uses methods from dynamics to show that the braid group cannot be realized by diffeomorphisms. As a corollary we obtain a new and relatively simple proof of Morita’s result in the optimal genus range.