Local and nonlocal Poincar\'e inequalities on Lie groups

We prove a local $L^p$-Poincar\'e inequality, $1\leq p < \infty$, on noncompact Lie groups endowed with a sub-Riemannian structure. We show that the constant involved grows at most exponentially with respect to the radius of the ball, and that if the group is nondoubling, then its growth is indeed, in general, exponential. We also prove a nonlocal $L^2$-Poincar\'e inequality with respect to suitable finite measures on the group.