In this thesis we discuss a recent result on the finite presentation of the tame fundamental group.
Remember that the étale fundamental group is a profinite group governing the study of finite étale covers of a fixed algebraic variety; in the case of an open variety, however, this group turns out to be too big and we prefer to work with a variant, called the tame fundamental group, which prescribes the ramification "at infinity".
Then, given a smooth quasi-projective variety over an algebraically closed field of positive characteristic, under the somewhat technical assumption that it admits a good compactification, the main result of our thesis is to show that its tame fundamental group is topologically finitely presented.
In particular, if the variety is projective, we know that its étale fundamental group is finitely presented, which in positive characteristic is already a novel result.