  We show that every graded nilpotent Lie group $G$ of step $r$, equipped with
a left invariant metric homogeneous with respect to the dilations induced by
the grading, (this includes all Carnot groups with Carnot-Caratheodory metric)
is Markov $p$-convex for all $p \in [2r,\infty)$. We also show that this is
sharp whenever $G$ is a Carnot group with $r \leq 3$, a free Carnot group, or a
jet space group; such groups are not Markov $p$-convex for any $p \in (0,2r)$.
This continues a line of research started by Li who proved this sharp result
when $G$ is the Heisenberg group. As corollaries, we obtain new estimates on
the non-biLipschitz embeddability of some finitely generated nilpotent groups
into nilpotent Lie groups of lower step. Sharp estimates of this type are known
when the domain is the Heisenberg group and the target is a uniformly convex
Banach space or $L^1$, but not when the target is a nonabelian nilpotent group.
