We prove that strongly continuous groups generated by first-order systems $D$ on Riemannian manifolds have finite propagation speed. The new direct proof for self-adjoint systems also provides a new approach to the weak Huygens' principle for second-order hyperbolic equations. The techniques are also combined with the resolvent approach to sectorial operators to obtain $L^2$ off-diagonal estimates for functions of $D$, which are the starting point for obtaining $L^p$ estimates for Riesz transforms on manifolds where the heat semigroup does not satisfy pointwise Guassian bounds. The two approaches are then combined via a Calder\'{o}n reproducing formula that allows for the analysing function to interact with $D$ through the holomorphic functional calculus whilst the synthesising function interacts with $D$ through the Fourier transform. This is joint work with P.~Auscher and A.~McIntosh.
