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

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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