Fixed points, local monodromy, and incompressibility of congruence covers

TL;DR

This paper establishes a fixed point theorem for local monodromy actions on étale covers, providing geometric proofs for incompressibility of Shimura varieties and moduli spaces, extending results to exceptional groups and reductions modulo primes.

Contribution

It introduces a geometric approach to prove incompressibility results, broadening the scope to include exceptional groups and reductions modulo primes.

Findings

Fixed point theorem for local monodromy groups

Lower bounds in essential dimension derived

Incompressibility results for Shimura varieties extended

Abstract

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.