Currents, Systoles, and Compactifications of Character Varieties

We study the Thurston–Parreau boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bi-Lipschitz equivalent to the length function with respect to a hyperbolic metric. The results of this paper on currents generalise the ones in arXiv:1710.07060v1 to the case of surfaces of finite area with geodesic boundary. Concerning the Thurston–Parreau boundary we improve on the results in arXiv:1710.07060v1 by showing that for higher rank groups, said open set of discontinuity is not empty. We give also explicit examples in the case of the \SL(3,\mathbb R)-Hitchin component.