Iterated Monodromy Groups of Rational Mappings

Abstract

We give the wreath recursion presentations of iterated monodromy groups of post-critically finite quadratic rational mappings fc whose ramification portrait are of the form 0 ↦ av2 ↦ … ↦ avm ↦1 ↦ ∞ ⇌ 1 To find a pattern of these wreath recursions, we compute the wreath recursions of iterated monodromy groups of capture maps composed with the Basilica polynomial. This computation gives rise to the notion of addresses, which is used to represent wreath recursions. Then we conjecture that each capture map composed with the Basilica polynomial is topologically equivalent to a post-critically finite quadratic rational mapping, and thus we conclude that the iterated monodromy groups of fc can be represented by addresses.