  We show that in an $m$-step Carnot group, a probability measure with finite
$m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a
polynomial in the moments of the measure, with respect to exponential
coordinates. Using this, we improve the main technical result of our previous
paper concerning Sobolev mappings between Carnot groups; as a consequence, a
number of rigidity and structural results from recent papers hold under weaker
assumptions on the Sobolev exponent. We also give applications to quasiregular
mappings, extending earlier work in the $2$-step case to general Carnot groups.
