## Dynamics of $$\text{Out}(F_n)$$ on the boundary of outer space.(English)Zbl 1045.20034

The outer automorphism group $$\text{Out}(F_n)$$ of the free group $$F_n$$ of rank $$n$$ acts properly discontinuously on outer space $$CV_n$$ which is a contractible space introduced by M. Culler and K. Vogtman in analogy with Teichmüller space and the action of the mapping class or modular group [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]; it is the set of minimal free isometric actions of $$F_n$$ on simplicial $$\mathbb{R}$$-trees modulo equivariant homothety and has a natural compactification in the set of minimal isometric actions of $$F_n$$ on $$\mathbb{R}$$-trees.
“In this paper, we study the dynamics of the action of $$\text{Out}(F_n)$$ on the boundary $$\partial CV_n$$ of outer space: we describe a proper closed $$\text{Out}(F_n)$$-invariant subset $${\mathcal F}_n$$ of $$\partial CV_n$$ such that $$\text{Out}(F_n)$$ acts properly discontinuously on the complementary open set. Moreover, we prove that there is precisely one minimal non-empty closed invariant subset $${\mathcal M}_n$$ in $${\mathcal F}_n$$. This set $${\mathcal M}_n$$ is the closure of the $$\text{Out}(F_n)$$-orbit of any simplicial action lying in $${\mathcal F}_n$$. We also prove that $${\mathcal M}_n$$ contains every action having at most $$n-1$$ ergodic measures. This makes us suspect that $${\mathcal M}_n={\mathcal F}_n$$. Thus $${\mathcal F}_n$$ would be the limit set of $$\text{Out}(F_n)$$, the complement of $${\mathcal F}_n$$ being its set of discontinuity.”

### MSC:

 20F65 Geometric group theory 20E08 Groups acting on trees 57M07 Topological methods in group theory 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$

Zbl 0589.20022
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