Schottky groups, Mumford curves, and their Berkovich analytification
17th May 2023, 2:30 pm – 3:30 pm
Fry Building, 2.04
Let k be a valued field. A Schottky group is a free, finitely generated subgroup of PGL_2(k) whose action on the projective line enjoys good topological and dynamical properties.
The theory of Schottky groups sits at the intersection of algebra and geometry and it is rich of applications to geometric group theory, algebraic geometry, dyanamical systems, and number theory.
If k is the field of complex numbers, Schottky groups acquired a central role already in the 1910’s, thanks to Koebe’s theory of uniformization of Riemann surfaces (i.e. complex algebraic curves). If k is non-archimedean (e.g. C((t)) or the field of p-adic numbers) Mumford proved that these groups provide a nice theory of uniformization for certain algebraic curves, now called Mumford curves.
In this talk, I will present two results on Mumford curves. The first (joint with Jérôme Poineau) exploits the theory of Berkovich spaces to build a 'universal’ Mumford curve, encoding in the same object the complex and non-archimedean theories of uniformization; the second (joint with Andrew Obus) explains the behavior of curves that become Mumford curves after base-change, allowing us to answer open questions about the minimal extension yielding semi-stable reduction of these curves over a discrete valuation ring.