Restrictions on the prime to $p$ fundamental group of a smooth projective variety

Abstract

The goal of this paper is to obtain restrictions on the prime to p quotient of the \'etale fundamental group of a smooth projective variety in characteristic $p\ge 0$. The results are analogues some theorems in the study of K\"ahler groups. Our first main result is that such groups are indecomposable under coproduct. The second gives a classification of the pro-$\ell$ parts of one relator groups in this class.