We will describe a method of determining when two Thurston maps (post-critically finite topological polynomials) are combinatorially equivalent. In particular, we will answer questions posed by J. Hubbard and A. Douady about compositions of Dehn twists with z 2 +i and with the “rabbit”. On our way we will get many exotic groups: an uncountable family of “indistinguishable” groups, groups of intermediate growth, exotic amenable groups, etc.