Room 515, Cosmology Building, NTU
(臺灣大學次震宇宙館 515研討室)
An Embedding of the Arboreal Galois Group for PCF Maps
Jun-Wen Peng (University of Rochester)
Abstract:
Let
![](https://chart.googleapis.com/chart?cht=tx&chl=%24f%3A%5Cmathbb%7BP%7D%5E1_K%5Cto%5Cmathbb%7BP%7D%5E1_K%24&chf=bg,s,333333&chco=ffffff)
be a rational map defined over a field
![](https://chart.googleapis.com/chart?cht=tx&chl=%24K%24&chf=bg,s,333333&chco=ffffff)
, and let
![](https://chart.googleapis.com/chart?cht=tx&chl=%24K_%7Bn%7D%24&chf=bg,s,333333&chco=ffffff)
be the splitting field of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24f%5En(X)-%5Calpha%20%3D%200%20%24&chf=bg,s,333333&chco=ffffff)
where
![](https://chart.googleapis.com/chart?cht=tx&chl=%24f%5En%24&chf=bg,s,333333&chco=ffffff)
is the n-th iterate of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24f%24&chf=bg,s,333333&chco=ffffff)
. We study the Galois group
![](https://chart.googleapis.com/chart?cht=tx&chl=%24G_n%3DGal(K_n%2FK)%24&chf=bg,s,333333&chco=ffffff)
. Odoni has showed that, avoiding a finite subset of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cmathbb%7BP%7D%5E1_K%24&chf=bg,s,333333&chco=ffffff)
, the profinite group $G_\infty = \varprojlim_n G_n$ acts on the infinite
![](https://chart.googleapis.com/chart?cht=tx&chl=%24d%24&chf=bg,s,333333&chco=ffffff)
-ary regular tree
![](https://chart.googleapis.com/chart?cht=tx&chl=%24T_%5Cinfty%24&chf=bg,s,333333&chco=ffffff)
, and hence we obtain a Galois representative, so called arboreal representative, by embedding
![](https://chart.googleapis.com/chart?cht=tx&chl=%24G_%5Cinfty%24&chf=bg,s,333333&chco=ffffff)
to the automorphism of the tree Aut
![](https://chart.googleapis.com/chart?cht=tx&chl=%24(T_%5Cinfty)%24&chf=bg,s,333333&chco=ffffff)
.
Generically, this embedding is surjective. However when
![](https://chart.googleapis.com/chart?cht=tx&chl=%24f%24&chf=bg,s,333333&chco=ffffff)
is a post-critical-finite(PCF) map, Jones showed that the image of
![](https://chart.googleapis.com/chart?cht=tx&chl=%24G_%5Cinfty%24&chf=bg,s,333333&chco=ffffff)
is an infinite index subgroup of Aut
![](https://chart.googleapis.com/chart?cht=tx&chl=%24(T_%5Cinfty)%24&chf=bg,s,333333&chco=ffffff)
. By explicitly computing the discriminant of a PCF map, we are able to find two kinds of infinite index subgroups of Aut
![](https://chart.googleapis.com/chart?cht=tx&chl=%24(T_%5Cinfty)%24&chf=bg,s,333333&chco=ffffff)
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.