2007: An Arboreal Odyssey: A View of Arboreal Galois Representations and Applications, from Early in the Subject's History

TL;DR

This 2007 survey provides a historical overview of arboreal Galois representations, highlighting their development from the 1980s and early 2000s, and discusses their applications and recent references.

Contribution

It offers an informal, historical perspective on the early development and renewed interest in arboreal Galois representations, serving as a reference for subsequent research.

Findings

Historical context of arboreal Galois representations from the 1980s to 2007

Connections to recent work and references in the field

Highlights of applications and developments in the subject

Abstract

The study of arboreal Galois representations (that is, Galois groups arising from iteration of polynomial and rational functions) originated with work of Odoni in the 1980s. Beginning in the early 2000s it underwent a period of renewed interest, which continues to this day. Written in 2007, this survey article gives a sense of the subject from the early days of this renewal. It is presented here as a document of historical interest -- precisely as originally written -- and because some recent work has referenced specific pieces of it. It was written as an informal document, and not intended to be published. Much, though not all, of the content overlaps with the 2013 survey article ``Galois representations from pre-image trees: an arboreal survey" of the author.