Exponential growth of some iterated monodromy groups

TL;DR

This paper demonstrates that certain iterated monodromy groups associated with complex rational maps exhibit exponential growth, expanding understanding beyond polynomial cases and including maps with diverse Julia set structures.

Contribution

It proves exponential growth for the iterated monodromy groups of several non-polynomial rational maps, including those with complex Julia sets and obstructed Thurston maps, and introduces a non-renormalizable polynomial with exponential growth.

Findings

Exponential growth shown for non-polynomial rational maps' IMGs.

First example of a non-renormalizable polynomial with exponential IMG growth.

Includes maps with Julia sets as the whole sphere and Sierpiński carpet.

Abstract

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpi\'{n}ski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose $\operatorname{IMG}$ has exponential growth.