This is a sequel to a paper by S. Parsad et al. [J. Topol. Anal. 11, No. 4, 929–964 (2019; Zbl 1432.57047)] where the authors presented a structure theorem for the realization of a finite order mapping class of a surface \(S_g\) as an isometry of a hyperbolic structure on \(S_g\) (in the sense of the Nielsen realization problem; as basic building blocks they consider periodic mapping classes which can be realized by an embedding of the surface into Euclidean space \(\mathbb R^3\) invariant under a rotation of \(\mathbb R^3\), or as a rotation of a polygon with a suitable side-pairing).
In the present paper the authors elaborate on this, in particular they give a description of the fixed point set in Teichmüller space of a periodic mapping class (whose dimension is the dimension of the Teichmüller space of the quotient orbifold of \(S_g\) by a periodic realization of the mapping class). As a Corollary, the fixed point set of an arbitrary finite group of mapping classes (containing an element of order at least 3) is not symplectomorphic to the Euclidean space of the same dimension. The authors give some other applications, most notably they indicate a topological proof of the Nielsen realization problem for cyclic groups of mapping classes (since Nielsen’s original proof from 1942 has a gap, the first proof for cyclic and solvable groups is by Fenchel 1948 using Teichmüller theory; the first purely topological proof for cyclic and solvable groups (along the lines of Nielsen’s original approach as generalized by Zieschang) is given in a paper by B. Zimmermann [Math. Ann. 229, 279–288 (1977; Zbl 0342.20025)], the first proof for arbitrary finite groups by Kerckhoff 1983 (again based on Teichmüller theory), the first purely topological proof (along Nielsen’s lines) for arbitrary finite groups by D. Gabai [Ann. Math. (2) 136, No. 3, 447–510 (1992; Zbl 0785.57004)]).