The arithmetic basilica: A quadratic PCF arboreal Galois group

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 x(0) is an element of 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) = z(2) - 1 when the field K and root point x(0) 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 x(0). (C) 2021 Elsevier Inc. All rights reserved.