A just-infinite iterated monodromy group without the congruence subgroup property

TL;DR

This paper constructs a specific polynomial's iterated monodromy group that is just-infinite, regular branch, and lacks the congruence subgroup property, providing a novel example in the field.

Contribution

It presents the first example of a polynomial's iterated monodromy group with these combined properties, expanding understanding of such groups.

Findings

The group is just-infinite and regular branch.

It does not have the congruence subgroup property.

Details about the congruence, rigid, and branch kernels are provided.

Abstract

We prove that the iterated monodromy group of the polynomial $z^2+i$ is just-infinite, regular branch and does not have the congruence subgroup property. This yields the first example of an iterated monodromy group of a polynomial with these properties. Additional information is provided about the congruence kernel, rigid kernel and branch kernel of this group.