Arithmetic iterated monodromy groups of postcritically finite quadratic polynomials with periodic postcritical orbits are densely settled.
Szamuely,Galois Groups and Fundamental Groups
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Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials
Arithmetic iterated monodromy groups of postcritically finite quadratic polynomials with periodic postcritical orbits are densely settled.