A large arboreal Galois representation for a cubic postcritically finite polynomial

TL;DR

This paper describes the Galois representation associated with a specific cubic polynomial, revealing new examples outside classical families and analyzing the structure and density properties of related prime divisors and convergence sets.

Contribution

It provides the first example of a non-classical cubic polynomial with a detailed arboreal Galois representation and analyzes its Hausdorff dimension and prime density results.

Findings

Galois action has Hausdorff dimension between cyclic and symmetric wreath products.

Zero-density result for prime divisors in polynomial orbits.

Zero-density result for convergence places of Newton's method.

Abstract

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.