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We give a complete description of the arboreal Galois representation of acertain postcritically finite cubic polynomial over a large class of numberfields and for a large class of basepoints. This is the first such example thatis not conjugate to a power map, Chebyshev polynomial, or Latt\`es map. Theassociated Galois action on an infinite ternary rooted tree has Hausdorffdimension bounded strictly between that of the infinite wreath product ofcyclic groups and that of the infinite wreath product of symmetric groups. Wededuce a zero-density result for prime divisors in an orbit under thispolynomial. We also obtain a zero-density result for the set of places ofconvergence of Newton's method for a certain cubic polynomial, thus resolvingthe first nontrivial case of a conjecture of Faber and Voloch.

A large arboreal Galois representation for a cubic postcritically finite polynomial