A large arboreal Galois representation for a cubic postcritically finite polynomial

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 Lattès map....

Full description

Saved in:
Bibliographic Details
Published in:Research in number theory 2017-12, Vol.3 (1), p.1-21
Main Authors: Benedetto, Robert L., Faber, Xander, Hutz, Benjamin, Juul, Jamie, Yasufuku, Yu
Format: Article
Language:English
Subjects:
Chebyshev approximation
Mathematics
Mathematics and Statistics
Number Theory
Polynomials
Representations
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 Lattès 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.
ISSN:2363-9555
DOI:10.1007/s40993-017-0092-8