Speaker : Wayne Jun-Wen Peng (University of Rochester)
Title : Recent development in the arboreal Galois representatives for PCF maps over a global field of characteristic 0
Time : 2021-05-07 (Fri) 13:30 - 14:45
Place : Conference Room, Institute of Mathematics (NTU Campus)
Abstract: Let f be a rational function defined over a global field $K$ of characteristic $0$. The critical points, zeros of the derivative of $f$, determine many dynamical properties of $f$. A rational function $f$ is post-critical finite (PCF) if all critical points of f are preperiodic, i.e. $\{f^n(c)\mid n\in\mathbb{N}\text{ and }f'(c)=0\}$, where $f^n$ is the n-th iterate of $f$, is a finite set. Let $d$ be the degree of $f$. Let $t$ be in $K$ or transcendental over $K$. One can embed the n-th dynamical Galois group, the Galois group of the splitting field of $f^n(X)-t$ over $K(t)$, into the automorphism group $\text{Aut}(T_n^d)$ where $T_n^d$ is a d-ary regular rooted tree up to n levels, and it was known that the index of the image of the embedding is unbounded as $n$ goes to infinity if $f$ is PCF.
In this talk, we will define two kinds of subgroups, denoted by $E_n^d$ and $F_n^d$, of $Aut(T_n^d)$ such that $E_n^d$ and $F_n^d$ has unbounded index in $\text{Aut}(T_n^d)$, and the embedding for a PCF map is a subgroup of $E_n^d$ or $F_n^d$. A degree $d$ PCF map $B_d$ is known as a 0-cycle Belyi map if $B_d$ has three and only three critical points which are fixed by $B_d$. We will talk about one recent result showing that the $n$-th dynamical Galois group of a 0-cycle Belyi map $B_d$ with only two exceptions is isomorphic to $E_n^d$ or the $n$-fold wreath product of alternative group $A_d$ if $t$ is transcendental. We will give a degree 3 not 0-cycle Belyi map that has the $n$-th dynamical Galois group isomorphic to $E_n^3$.
If we have time, we will discuss a possible strategy to attack some new family of PCF maps, and talk a little bit about an application of the arboreal Galois representatives in the number theory.