Blocks of monodromy groups in Complex Dynamics

TL;DR

This paper investigates the block structure of monodromy groups in complex dynamics, showing that for many polynomials, the structure is simple, but exceptions exist, notably degree 6 polynomials, which have unique properties.

Contribution

It demonstrates the conditions under which monodromy groups have trivial block structures and provides a specific counterexample at degree 6, addressing open problems in the field.

Findings

No large blocks for prime power degree polynomials

Constructed a degree 6 polynomial with non-trivial block structure

Degree 6 is the lowest degree where monodromy group isn't determined by post-critical set

Abstract

Motivated by a problem in complex dynamics, we examine the block structure of the natural action of monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.