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Nielsen realization for infinite-type surfaces. (English) Zbl 1468.57011

Let \(S\) be an orientable surface of infinite type or a hyperbolic surface of finite type. Let \(\mathrm{Map}(S)\) be the mapping class group of \(S\) and let \(\mathcal{T}(S)\) be the Teichmüller space of \(S\). When \(S\) of finite type, the Nielsen realization problem [J. Nielsen, Acta Math. 75, 23–115 (1942; Zbl 0027.26601)] asked whether a finite subgroup \(G < \mathrm{Map}(S)\) can be realized as a group of isometries of some hyperbolic metric on \(S\). Nielsen himself showed [loc. cit.] the positivity of the result for the case when \(G\) is cyclic, which was later extended by W. Fenchel [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 5, 326–329 (1948; Zbl 0036.12902); Mat. Tidsskr. B 1950, 90–95 (1950; Zbl 0039.19303)] to the case when \(G\) is solvable. However, the general case remained open for a few decades until it was finally answered in the affirmative by S. P. Kerckhoff [Ann. Math. (2) 117, 235–265 (1983; Zbl 0528.57008)]. Kerckhoff used Thurston’s earthquake deformations to show that the natural action of \(G\) on \(\mathcal{T}(S)\) has a fixed point.
The study of the mapping class groups of infinite-type surfaces \(S\) (also known as big mapping class groups) has garnered a lot of interest [K. Ohshika (ed.) and A. Papadopoulos (ed.), In the tradition of Thurston. Geometry and topology (to appear). Cham: Springer (2020; Zbl 1470.57002), Chapter 12] in recent years. A natural question that arises in this context is whether one could derive an analog of the Nielsen realization theorem for an infinite-type surface \(S\). In a recent paper [T. Aougab et al., “Isometry groups of infinite-genus hyperbolic surfaces”, Preprint, arXiv:2007.01982, to appear in Math. Ann.], it has been shown that such a result does hold true for the countable subgroups of the mapping class group for several infinite-type surfaces.
In this paper, the main result extends the Nielsen realization theorem to orientable surfaces of infinite type. A key idea in Kerckhoff’s proof lies in showing that the length function \(\ell_{G} : \mathcal{T}(S) \to \mathbb{R}^+\) (that sends a hyperbolic structure to the sum of the geodesic lengths of a finite \(G\)-invariant collection of filling curves on \(S\)) attains a unique minimum. But this argument does not naturally generalize to the infinite-type setting where such a length function would diverge. However, this paper adapts an alternative approach by showing the existence of an exhaustion \(S_0 \subset S_1 \subset \ldots\) of connected finite-type \(G\)-invariant hyperbolic subsurfaces of \(S\). The result is then derived by inductively applying the Nielsen realization theorem to each piece \(S_k \setminus S_{k-1}\), and then carefully assembling the resultant structures in the pieces to yield a hyperbolic structure on \(S\) that realizes \(G\) as an isometry group.
Let \(S = \mathbb{R}^2 \setminus K\), where \(K\) is the Cantor set and let \(S^1_C\) be the conical circle comprising geodesics originating at \(\infty\). By analyzing the rotation numbers associated with the action of the torsion elements of \(\mathrm{Map}(S)\) on \(S^1_C\), the following application of the main result is derived.
Theorem. Finite-order elements of \(\mathrm{Map}(S)\) fix at most one point in \(K\). Moreover, for \(n \geq 2\), the torsion elements in \(\mathrm{Map}(S)\) of order \(n\) form at most \(2 \varphi(n)\) conjugacy classes, where \(\varphi\) denotes the Euler totient function.
Furthermore, for an arbitrary orientable surface \(S\) of infinite type, the authors show the existence of an open neighborhood of the identity in \(\mathrm{Map}(S)\) that contains no nontrivial torsion elements. This leads to another application of the main result.
Theorem. A topological group containing a sequence of nontrivial finite-order elements limiting to the identity cannot imbed in \(\mathrm{Map}(S)\).
As a final application, by proving that a compact subgroup of \(\mathrm{Map}(S)\) is virtually torsion-free consisting only of finite-order elements, they establish the following.
Theorem. The compact subgroups of \(\mathrm{Map}(S)\) are finite, and the locally compact subgroups of \(\mathrm{Map}(S)\) are discrete.

MSC:

57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
57M50 General geometric structures on low-dimensional manifolds
20E34 General structure theorems for groups
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References:

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