  We give a complete description of the arboreal Galois representation of a
certain postcritically finite cubic polynomial over a large class of number
fields and for a large class of basepoints. This is the first such example that
is not conjugate to a power map, Chebyshev polynomial, or Latt\`es map. The
associated Galois action on an infinite ternary rooted tree has Hausdorff
dimension bounded strictly between that of the infinite wreath product of
cyclic groups and that of the infinite wreath product of symmetric groups. We
deduce a zero-density result for prime divisors in an orbit under this
polynomial. We also obtain a zero-density result for the set of places of
convergence of Newton's method for a certain cubic polynomial, thus resolving
the first nontrivial case of a conjecture of Faber and Voloch.
