  Suppose A is an open subset of a Carnot group G, where G has a discrete
analogue, and H is another Carnot group. We show that a Lipschitz function from
A to H whose image has positive Hausdorff measure in the appropriate dimension
is biLipschitz on a subset of A of positive Hausdorff measure. We then
construct Lipschitz maps from open sets in Carnot groups to Euclidean space
that do not decrease dimension. Finally, we discuss two counterexamples to
explain why Carnot group structure is necessary for these results.
