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 --> $z^2-1$.