Nilpotent pseudogroups of functions on an interval

Abstract.A near-identity nilpotent pseudogroup of order m ≥ 1 is a family f1, . . . , fn : (-1, 1) → ℝ of C2 functions for which: $$ {\left| {f_{i} - {\text{id}}} \right|}_{{C^{1} }} < \in $$ for some small positive real number ∈ < 1/10m+1 and commutators of the functions fi of order at least m equal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds.