We are interested in finite generation or finite presentation of fundamental groups as topological profinite groups. Our knowledge of group theoretic properties of étale fundamental groups relies traditionally on Riemann's existence theorem (in char 0) and Grothendieck's specialization map (for the transition to char p). But not all varieties lift to characteristic 0. Building on recent results by Esnault, Shusterman and Srinivas for smooth projective varieties in char p, we are going to explain in the talk how to generalize finite presentation to arbitrary proper varieties (joint work with Lara and Srinivas). Furthermore, we introduce an adic tameness condition and discuss finite generation/presentation of tame fundamental groups for rigid analytic spaces. The second part is joint work with Achinger, Lara and Hübner.
Jakob Stix (Goethe-Universität Frankfurt)
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