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Elliptic actions on Teichmüller space. (English) Zbl 1427.57009

Let \(S\) be a surface of finite topological type, let \(\mathcal{MCG}(S)\) be the mapping class group of \(S\), that acts on the Teichmüller space \(\mathcal{T}(S)\). The solution of the Nielsen Realization Problem – see [S. P. Kerckhoff, Ann. Math. (2) 117, 235–265 (1983; Zbl 0528.57008)] for the case of a closed surface – asserts that any finite subgroup \(H<\mathcal{MCG}(S)\) has a non-empty set of fixed points \(\text{Fix}(H)\). This is true for the action of a finite-group on any simply-connected complete manifold that is negatively curved, by an argument of Cartan that uses the notion of a {barycenter} of an orbit – see for example, Chapter I, Theorem 13.5 of [S. Helgason, Differential geometry, Lie groups, and symmetric spaces, reprint with corrections of the 1978 original (2001; Zbl 0993.53002)]. However, the Teichmüller metric \(d_T\) on \(\mathcal{T}(S)\) is not negatively curved, and the Weil-Petersson metric \(d_{WP}\) on \(\mathcal{T}(S)\), though negatively curved, is incomplete, which makes Kerckhoff’s result striking.
This paper studies the structure of the set of \(R\)-almost fixed points in \(\mathcal{T}(S)\), that are moved by the action of the finite subgroup \(H\) by a distance bounded by \(R>0\) in the Teichmüller metric: \[ \text{Fix}_R^T(H) = \{X \in \mathcal{T}(S)\mid \text{diam}_T(H\cdot X) <R\}. \]
The main result of the paper is that \(\text{Fix}_R^T(H)\) is \(R^\prime\)-close to the fixed-point set \(\text{Fix}(H)\), for some \(R^\prime>0\) that only depends on \(R\) and the surface \(S\) (see Theorem 1.2 or 5.6).
A key ingredient of the proof is a coarse distance formula in the augmented marking complex \(\mathcal{AM}(S)\), that the author had shown is a quasi-isometric model of \((\mathcal{T}(S), d_T)\), in previous work – see [J. Lond. Math. Soc., II. Ser. 94, No. 3, 933–969 (2016; Zbl 1360.30038)]. The proof also uses extensions of technical results arising from the “hierarchy machinery” in [H. A. Masur and Y. N. Minsky, Geom. Funct. Anal. 10, No. 4, 902–974 (2000; Zbl 0972.32011)], and [J. Tao, ibid. 23, No. 1, 415–466 (2013; Zbl 1286.57020)], to the setting of \(\mathcal{AM}(S)\).
The work of J. Tao, together with the main result, is used to prove the existence of “coarse barycenters” of the orbit of any finite-order element of \(\mathcal{MCG}(S)\) (Theorem 1.3 or 6.1). Finally, in contrast to these results reminiscent of negative curvature, the author gives examples where \(\text{Fix}_R^T(H)\) is not quasiconvex (for some \(R>0\), some surface \(S\) and finite subgroup \(H<\mathcal{MCG}(S)\)) by adapting constructions in [K. Rafi, Geom. Topol. 18, No. 5, 3025–3053 (2014; Zbl 1314.30082)]. The author concludes with a discussion of how these examples are, in fact, expected to be part of a common phenomenon.
The paper is very well-written, and the technical arguments are well motivated.

MSC:

57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
57M50 General geometric structures on low-dimensional manifolds
20F65 Geometric group theory
30F60 Teichmüller theory for Riemann surfaces
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References:

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