Iterated monodromy groups of critically fixed (anti-)rational maps

Abstract

Iterated monodromy groups (in short, IMGs) are self-similar groups naturally associated to iterations of (anti-)rational maps on the Riemann sphere. In this thesis, we study the properties of the IMGs of critically fixed (anti-)rational maps; critically fixed maps being those maps whose critical points are also fixed points. More specifically, we prove that the IMGs of critically fixed (anti-)polynomials are regular branch on the subgroup of group elements with even permutational part. In the case of polynomials, we make use of the one-to-one correspondence between the conformal conjugacy classes of critically fixed polynomials and the isomorphism classes of connected planar embedded graphs. Similarly, in the case of anti-polynomials, we use that there is a one-to-one correspondence between the conformal conjugacy classes of critically fixed anti-rational maps and the isomorphism classes of unobstructed topological Tischler graphs. Not being able to prove a similar statement in the general case of (anti-)rational maps, we discuss some motivating examples and explain some of the difficulties.