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

Published online by Cambridge University Press:  20 April 2007

ILYA KAPOVICH
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA (e-mail: kapovich@math.uiuc.edu)
MARTIN LUSTIG
Affiliation:
Mathématiques (LATP), Université Paul Cézanne–Aix Marseille III, av. E. Normandie-Niémen, 13397 Marseille 20, France (e-mail: martin.lustig@univ-cezanne.fr)

Abstract

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 $\overline{CV(F_k)}=CV(F_k)\cup \partial CV(F_k)$.

Type
Research Article
Copyright
2007 Cambridge University Press

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