Fixed points, local monodromy, and incompressibility of congruence covers

We prove a fixed point theorem for the action of certain local monodromy groups on étale covers and use it to deduce lower bounds on essential dimension. In particular, we give more geometric proofs of some of the results of a paper of Farb, Kisin and Wolfson, which uses arithmetic methods to prove incompressibility results for Shimura varieties and moduli spaces of curves. Our method allows us to prove new results for exceptional groups, applies also to the reduction modulo good primes of congruence covers of Shimura varieties and moduli spaces of curves, and also to certain “quantum” covers of moduli spaces of curves arising from a certain TQFT.