This is a report on joint work with Hélène Esnault. Let X be a smooth projective variety over the complex numbers C. Let M be the moduli space of irreducible representations of the topological fundamental group of X of a fixed rank r. Then M is a finite type scheme over the spectrum of the integers Z. We may ask whether M is pure over Z in the sense of Raynaud-Gruson, for example we can ask if the irreducible components of M which dominate Spec(Z) actually surject onto Spec(Z). We will explain what this means, present a weak answer to this question, apply this to exclude some abstract groups as the fundamental groups of smooth projective varieties over C, and we discuss what other phenomena can be studied using the method of proof.
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