The actions of $Out(F_k)$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility

Abstract

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb PCurr(F_k)$ that is either $Out(F_k)$-equivariant or $Out(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the "length-function" boundary of Culler-Vogtmann's Outer space $CV(F_k)$, and $\mathbb PCurr(F_k)$ is the space of projectivized geodesic currents for $F_{k}$. We also prove that, if $k\ge 3$, for the action of $Out(F_k)$ on $\mathbb PCurr(F_{k})$ and for the diagonal action of $Out(F_k)$ on the product space $\partial CV(F_k)\times \mathbb PCurr(F_k)$ there exist unique non-empty minimal closed $Out(F_k)$-invariant sets. Our results imply that for $k\ge 3$ any continuous $Out(F_k)$-equivariant embedding of $CV(F_k)$ into $\mathbb PCurr(F_k)$ (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual "length-function" compactification $\bar{CV(F_k)}=CV(F_k)\cup \partial CV(F_k)$.