Subset Currents on Surfaces

Subset Currents on Surfaces Synopsis

"Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group 1() of a compact hyperbolic surface . Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on 1(), which we call subset currents on . We prove that the space SC() of subset currents on is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of 1(), each of which geometrically corresponds to a convex core of a covering space of . This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon's result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we e