  We prove a fixed point theorem for the action of certain local monodromy
groups on \'etale covers and use it to deduce lower bounds in essential
dimension. In particular, we give more geometric proofs of many (but not all)
of the results of the preprint 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 results for exceptional
groups, and also for the reduction modulo good primes of Shimura varieties and
moduli spaces of curves.
