we will discuss the properties of a 2-generator 2-relator group whichshares many properties with the famous richard thompson group f. thegroup arises as an ascending hnn extension of a certain group which isgenerated by a 3-state automaton. the latter group will be in the focusof our considerations. in particular we will explain that it is theiterated monodromy group of the map z --> $z^2-1$.