A universal nilpotent group of diffeomorphisms of the interval

Let N n be the group of n × n lower-triangular matrices with integer entries and ones in the diagonal. In this work we are interested in constructing a C 1 action on the interval of a specific and somewhat universal nilpotent group denoted by N, which is the union of all of them. This yields a unified version of a construction due to B. Farb and J. Franks.