In 1997, Shinichi Mochizuki proved that every outer automorphism of G_K, the absolute Galois group of a p-adic field K, preserving the ramification filtration, arises from a field isomorphism. On the other hand, there are many outer automorphisms of G_K which do not preserve the filtration, but our understanding of Out(G_K) remains limited, with few explicit automorphisms constructed.

Following the philosophy of arithmetic topology, pioneered by Barry Mazur, p-adic fields should be analogous to surfaces, suggesting that Out(G_K) should resemble the mapping class group of a surface.

In the talk I will explain some of the reasons for expecting such analogies. I will also outline some recent work of the speaker, in which, using the theory of graphs of groups and Bass–Serre theory, a generalization of Dehn twists was constructed. These give a family of infinite order outer automorphisms of GK(p), the maximal pro-p quotient of GK, which are “arithmetic Dehn twists”.

### Galois Seminar

Friday, April 26, 2024 - 3:30pm

#### Nadav Gropper

Penn & University of Haifa