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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
Software:
Bear
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References:
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