  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 $x_0\in K$,
analogous to the action of Galois on the $\ell$-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 $K=\mathbb{Q}$, our condition holds for infinitely many
choices of $x_0$.
