Exponential growth of some iterated monodromy groups

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpi\'{n}ski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose $\operatorname{IMG}$ has exponential growth.