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

• Autores: Ilya Kapovich, Martin Lustig
• Localización: Ergodic theory and dynamical systems, ISSN 0143-3857, Vol. 27, Nº 3, 2007, págs. 827-847
• Idioma: inglés
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• Resumen

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