Hurwitz Trees and Tropical Geometry

Abstract

The lifting problem in algebraic geometry asks when a finite group G acting on a curve defined over characteristic p > 0 lifts to characteristic 0. One object used in the study of this problem is the Hurwitz tree, which encodes the ramification data of a group action on a disk. In this thesis we explore the connection between Hurwitz trees and tropical geometry. That is, we can view the Hurwitz tree as a tropical curve. After exploring this connection we provide two examples to illustrate the connection, using objects in tropical geometry to demonstrate when a group action fails to lift.