  We prove the Sobolev-to-Lipschitz property for metric measure spaces
satisfying the quasi curvature dimension condition recently introduced in E.
Milman, The Quasi Curvature-Dimension Condition with applications to
sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear,
arXiv:1908.01513v5). We provide several applications to properties of the
corresponding heat semigroup. In particular, under the additional assumption of
infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for
the heat semigroup with respect to the distance, and prove the irreducibility
of the heat semigroup. These result apply in particular to large classes of
(ideal) sub-Riemannian manifolds.
