Some finiteness results for the étale fundamental group in positive characteristics

This talk will discuss some results on etale fundamental groups of varieties over an algebraically closed field of characteristic p > 0, based on joint work with Hélène Esnault and other coauthors. One result, along with Mark Schusterman, is that the tame fundamental group is finitely presented for such a variety which is the complement of an SNC divisor in a smooth projective variety. A second, along with Jakob Stix, is to give an obstruction for a smooth projective variety to admit a lifting to characteristic 0, in terms of the structure of its etale fundamental group as a profinite group. We will finally touch on some open questions.