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

TL;DR

This work proves that $\mathrm{IMG}(z^2+i)$ is just-infinite and regular branch while lacking the congruence subgroup property, providing the first such example among iterated monodromy groups of a quadratic polynomial. The approach centers on the maximal regular branching subgroup $K$ and an $L$-presentation, showing $K/K'\cong C_4^5$ and that the rigid kernel is trivial while the branch and congruence kernels are isomorphic to $C_4[[\partial T]]$. By analyzing abelianizations across levels and constructing the subgroups $\mathcal{K}$, the paper shows that $K'$ does not capture all level stabilizers, hence CSP fails. The results yield a detailed description of the profinite kernels (rigid, branch, and congruence) for this IMG and establish a concrete counterexample to the CSP within the class of just-infinite iterated monodromy groups.

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.

Figures (1)

• Figure 1: Automaton corresponding to $\mathop{\mathrm{IMG}}\nolimits(z^2+i)$.

Theorems & Definitions (24)

• Theorem A
• Theorem B
• Theorem C
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: Sunic2006
• Lemma 2.4
• proof