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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.

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
Full Text: DOI arXiv
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