Domains of proper discontinuity on the boundary of Outer space

TL;DR

This paper constructs domains of proper discontinuity for large subgroups of Out(F_N) acting on the boundary of Outer space and geodesic currents, extending previous work on mapping class groups.

Contribution

It introduces new domains of proper discontinuity for subgroups of Out(F_N) containing hyperbolic iwips, generalizing earlier results to Outer space.

Findings

Constructed domains of proper discontinuity in Outer space boundary.

Proved proper discontinuity of Out(F_N) action on currents with full support.

Extended McCarthy-Papadopoulos framework to Out(F_N).

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 "sufficiently large" subgroup of $Out(F_N)$ (that is, a subgroup containing a hyperbolic iwip). As a corollary we prove that for $N\ge 3$ the action of $Out(F_N)$ on the subset of $\mathbb PCurr(F_N)$ consisting of all projectivized currents with full support is properly discontinuous.