In this talk, sequel to the previous, we will construct the profinite iterated monodromy groups (pfIMGs) associated to rational functions, and describe an outer Galois action on the geometric pfIMG. This outer action factors through the constant field extension, an important invariant of the dynamical system. For polynomials, we show that the action is faithful after restricting to this field, and, moreover, that the cyclotomic character and an explicit group-theoretic property of the geometric pfIMG determine the constant field extension and arithmetic pfIMG completely.

This reduces us to calculating the geometric pfIMG and solving this problem. While the pfIMGs are constructed purely algebraically, a crucial topological input seems to be unavoidable, which we isolate and explain. Along the way, we will discuss analogies between dynamical systems and abelian varieties, some interesting coincidences and links to anabelian geometry, and general questions about outer actions on self-similar groups.

Speaker: Ophelia Adams, joint work with Trevor Hyde