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On dynamics of $$\mathrm{Out}(F_n)$$ on $$\mathrm{PSL}_2(\mathbb C)$$ characters. (English) Zbl 1282.57023
Let $$F$$ be a free group of rank at least two. $$\text{Out}(F)$$ naturally acts on the character variety of any Lie group $$G$$, and a natural problem is to understand the dynamic of that action.
In [W. Goldman, Geom. Funct. Anal. 17, No. 3, 793–805 (2007; Zbl 1139.57002)] it is proved that the action is ergodic for $$G=SU(2)$$, and conjectured that the same holds true for compact groups. In [T. Gelander, Isr. J. Math. 167, 15–26 (2008; Zbl 1171.22007)] the conjecture is positively answered.
The paper under review studies the case of $$G=\text{PSL}_2(\mathbb C)$$, which can be identified with the group of orientation preserving isometries of the hyperbolic space. In this case the character variety $$\chi(F)$$ admits a natural decomposition, up to a measure zero set, in discrete and dense representations. Moreover, the action of $$\text{Out}(F)$$ is not ergodic, as it acts properly discontinuously on the set of Schottky (discrete, faithful, and convex-cocompact) representations, which is the interior of the set of discrete representations. The main result of the paper implies that the action of $$\text{Out}(F)$$ is not ergodic even if restricted to the set of representations with dense image.
The author introduces the notion of primitive stable representations as those admitting a developing map from the Cayley graph of $$F$$ to the hyperbolic space such that axes of primitive elements are uniformly mapped to quasigeodesics. A first lemma is proved showing that the set of primitive stable representations is open and contains Schottky representations. Then it is proven that $$\text{Out}(F)$$ acts properly discontinuously on the set of primitive stable representations. Then, in Section 4 it is proved that the set of primitive stable representations contains some non-Schottky character. In particular, the domain of discontinuity of the action of $$\text{Out}(F)$$ on the $$\text{PSL}_2(\mathbb C)$$ character variety is strictly bigger than the set of discrete representations, hence the action of $$\text{Out}(F)$$ on the set of dense representations is not ergodic.
The proofs of Section 4 rely on a criterium of Whitehead for recognizing primitive elements and in the use of some beautiful (“a bit of” in the author’s words) hyperbolic geometry.

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) 20E36 Automorphisms of infinite groups 37A99 Ergodic theory
Bear
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