Domains of proper discontinuity on the boundary of Outer space

Abstract

Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any “dynamically large” subgroup of $\operatorname{Out}(F_N)$ (that is, a subgroup containing an atoroidal iwip).

As a corollary, we prove that for $N\ge3$ the action of $\operatorname{Out}(F_N)$ on the subset of $\mathbb{P}\operatorname{Curr}(F_N)$ consisting of all projectivized currents with full support is properly discontinuous.