Lecture Hall 5F, Cosmology Building, NTU

(臺灣大學宇宙學大樓 五樓演講廳)

An Embedding of the Arboreal Galois Group for PCF Maps

Jun-Wen Peng (University of Rochester)

Abstract:

Let

be a rational map defined over a field

, and let

be the splitting field of

where

is the n-th iterate of

. We study the Galois group

. Odoni has showed that, avoiding a finite subset of

, the profinite group $G_\infty = \varprojlim_n G_n$ acts on the infinite

-ary regular tree

, and hence we obtain a Galois representative, so called arboreal representative, by embedding

to the automorphism of the tree Aut

.

Generically, this embedding is surjective. However when

is a post-critical-finite(PCF) map, Jones showed that the image of

is an infinite index subgroup of Aut

. By explicitly computing the discriminant of a PCF map, we are able to find two kinds of infinite index subgroups of Aut

such that the arboreal Galois group of any PCF map can be embedded into one of them. People have found a family of PCF maps, called single-cycle Belyi map, of which the arboreal Galois groups are isomorphic to one of the subgroups. We are able to find a new PCF map that is also isomorphic to the subgroup.