The iterated monodromy group of z^2+i is just-infinite and fails the congruence subgroup property, with maximal branching subgroup abelianization C_4^5.
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A just-infinite iterated monodromy group without the congruence subgroup property
The iterated monodromy group of z^2+i is just-infinite and fails the congruence subgroup property, with maximal branching subgroup abelianization C_4^5.