The arithmetic basilica: A quadratic PCF arboreal Galois group

Abstract

The arboreal Galois group of a polynomial f over a field K encodes the action of Galois on the iterated preimages of a root point x0∈K, analogous to the action of Galois on the ℓ-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial f(z)=z2−1 when the field K and root point x0 satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of f. For K=Q, our condition holds for infinitely many choices of x0.