Finite Type Properties of Tame Fundamental Groups

Explore a comprehensive lecture on the finite type properties of fundamental groups in algebraic geometry. Delve into the study of finite generation and finite presentation of fundamental groups as topological profinite groups. Examine the traditional reliance on Riemann's existence theorem and Grothendieck's specialization map for understanding group theoretic properties of étale fundamental groups. Discover recent advancements in generalizing finite presentation to arbitrary proper varieties, based on work by Esnault, Shusterman, and Srinivas for smooth projective varieties in characteristic p. Learn about the introduction of an adic tameness condition and its implications for finite generation and presentation of tame fundamental groups in rigid analytic spaces. Gain insights from Jakob Stix's collaborative research with Lara, Srinivas, Achinger, and Hübner, expanding the understanding of fundamental groups in various mathematical contexts.