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We show that every graded nilpotent Lie group $G$ of step $r$, equipped witha left invariant metric homogeneous with respect to the dilations induced bythe 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 issharp whenever $G$ is a Carnot group with $r \leq 3$, a free Carnot group, or ajet 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 resultwhen $G$ is the Heisenberg group. As corollaries, we obtain new estimates onthe non-biLipschitz embeddability of some finitely generated nilpotent groupsinto nilpotent Lie groups of lower step. Sharp estimates of this type are knownwhen the domain is the Heisenberg group and the target is a uniformly convexBanach space or $L^1$, but not when the target is a nonabelian nilpotent group.

Estimates on the Markov Convexity of Carnot Groups and Quantitative Nonembeddability