Closure of orbits of the pure mapping class group in the character variety.

The pure mapping class group of an oriented surface acts on the character variety of the surface. We investigate the closure of its orbits under either Zariski or analytic topologies. We show that a generic infinite orbit is dense in an open subset of the corresponding modified relative character variety. Moreover, we specify that being generic can be captured as a combination of passing to a Zariski-open subscheme and avoiding a concrete set of exceptional cases. In addition, the functorial behavior of the involved schemes is described in detail. In particular, we answer a question posed by Goldman and place the recent work of Bourgain, Gamburd, and Sarnak on Markoff triples in a more general context.