Homology of jet groups

In this paper we compute the second homology of the discrete jet groups. Let R be the additive group of real numbers and R the multiplicative group of positive reals. The n jet group Jn = {rx+a2x + · · ·+anx | r ∈ R, ai ∈ R} is the group, under composition followed by truncation, of invertible, orientationpreserving real n-jets at 0. Consider the homomorphism D : Jn → R obtained by projecting onto the first coefficient, i.e. Df = first derivative of f at 0. Every jet with slope not equal to 1 is conjugate to its linear part. It follows there is a split exact sequence