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The actions of $$\text{Out}(F_k)$$ on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility. (English) Zbl 1127.20025
From the authors’ summary: We prove that for $$k\geq 5$$ there does not exist a continuous map $$\partial CV(F_k)\to\mathbb{P}\text{Curr}(F_k)$$ that is either $$\text{Out}(F_k)$$-equivariant or $$\text{Out}(F_k)$$-anti-equivariant. Here $$\partial CV(F_k)$$ is the ‘length function’ boundary of Culler-Vogtmann’s Outer space $$CV(F_k)$$, and $$\mathbb{P}\text{Curr}(F_k)$$ is the space of projectivized geodesic currents for $$F_k$$. We also prove that, if $$k\geq 3$$, for the action of $$\text{Out}(F_k)$$ on $$\mathbb{P}\text{Curr}(F_k)$$ and for the diagonal action of $$\text{Out}(F_k)$$ on the product space $$\partial CV(F_k)\times\mathbb{P}\text{Curr}(F_k)$$, there exist unique non-empty minimal closed $$\text{Out}(F_k)$$-invariant sets. Our results imply that for $$k\geq 3$$ any continuous $$\text{Out}(F_k)$$-equivariant embedding of $$CV(F_k)$$ into $$\mathbb{P}\text{Curr}(F_k)$$ (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual ‘length function’ compactification $$\overline{CV(F_k)}=CV(F_k)\cup\partial CV(F_k)$$.

MSC:
 20E36 Automorphisms of infinite groups 20F65 Geometric group theory 57M05 Fundamental group, presentations, free differential calculus 57M07 Topological methods in group theory 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
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