  We give lower bound estimates for the macroscopic scale of coarse
differentiability of Lipschitz maps from a Carnot group with the
Carnot-Carath\'{e}odory metric $(G,\dcc)$ to a few different classes of metric
spaces. Using this result, we derive lower bound estimates for quantitative
nonembeddability of Lipschitz embeddings of $G$ into a metric space $(X,d_X)$
if $X$ is either an Alexandrov space with nonpositive or nonnegative curvature,
a superreflexive Banach space, or another Carnot group that does not admit a
biLipschitz homomorphic embedding of $G$. For the same targets, we can further
give lower bound estimates for the biLipschitz distortion of every embedding $f
: B(n) \to X$, where B(n) is the ball of radius $n$ of a finitely generated
nonabelian torsion-free nilpotent group $G$. We also prove an analogue of
Bourgain's discretization theorem for Carnot groups and show that Carnot groups
have nontrivial Markov convexity. These give the first examples of metric
spaces that have nontrivial Markov convexity but cannot biLipschitzly embed
into Banach spaces of nontrivial Markov convexity.
