Artin characters, Hurwitz trees and the lifting problem

TL;DR

This paper investigates automorphism groups of the p-adic open disk, extending previous cyclic group results to all finite groups, and provides a counterexample related to the local lifting problem for quaternion groups.

Contribution

It generalizes results on automorphisms from cyclic to arbitrary finite groups and addresses the local lifting problem with a new counterexample.

Findings

Counterexample to the local lifting problem for generalized quaternion groups

Extension of automorphism group results to all finite groups

Advancement in understanding p-adic automorphisms

Abstract

We study finite groups of automorphisms of the $p$-adic open disk. In particular, we generalize results of Green, Matignon and Henrio from cyclic groups of order $p$ to arbitrary finite groups. As an application, we produce a counterexample to a question of Chinburg, Guralnick and Harbater, concerning the local lifting problem for generalized quaternion groups.