  We study $\mathbb{R}^k \times \mathbb{Z}^\ell$ actions on arbitrary compact
manifolds with a projectively dense set of Anosov elements and 1-dimensional
coarse Lyapunov foliations. Such actions are called totally Cartan actions. We
completely classify such actions as built from low-dimensional Anosov flows and
diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for
this class. This is achieved by introducing a new tool, the action of a
dynamically defined topological group describing paths in coarse Lyapunov
foliations, and understanding its generators and relations. We obtain
applications to the Zimmer program.
