On the rotation angles of a finite subgroup of a mapping class group. (English) Zbl 1178.30058

Let \(\Sigma^\sigma\) be a compact Riemann surface of genus \(\sigma\). Consider a subgroup \(G\) of the mapping class group of \(\Sigma^\sigma\). For any prime divisor \(p\) of order \(|G|\), there exists \(g\in G\) of order \(p\). Let \(q_1\), \(q_2,\ldots,q_b\) be fixed points of \(g\). Then \(g\) acts on the tangent space \(T_{q_j}\) as multiplication by \(\alpha^{t_j}\), \(1\leq t_j<p\). Here \(\alpha\) is the primitive \(p\)-th root of unity. Then \(\{t_1,t_2,\ldots, t_b\}\) is called the rotation angle of \(g\). Let \(Z_p\) be the cyclic group generated by \(g\). Denote by \(\tau\) the genus of \(\Sigma^\sigma/Z_p\). Then the Riemann-Hurwitz formula \(2\sigma-2=p(2\tau-2)+b(p-1)\) holds and \(\sum_{j=1}^b {t}^{-1}_j=0\). Here \({t}^{-1}_j\) is the inverse element of \(b_j\) in \(Z_p\). A rotation angle \(\{t_1,t_2,\ldots, t_b\}\) satisfying these conditions is called possible.
For integers \(1\leq z\), \(l<p\), denote by \(\Psi_p(z,l,t_1,t_2,\ldots, t_b)\) the number
\[ \frac{(p-1)(1-\sigma)(2l+1)}{2p} +\frac{1}{12 p}\sum_{i=1}^b \bigg( (p-1)(7p-11)zt_i+6 \sum_{j=m}^k f_p \Big(\Big[\frac{jp-1}{zt_i}\Big]-l-1\Big)\bigg), \]
where \(m=[(l+1)zt_i/p]+1\), \(k=[(l+p+1)zt_i/p]\), \(f_p=x^2-(p-2)x-(p-1)^2\), and \([x]\) is the entire part of \(x\). A rotation angle \(\{t_1,t_2,\ldots, t_b\}\) is called admissible if \(\Psi_p(z,l,t_1,t_2,\ldots, t_b)\) is an integer for any \(1\leq z\), \(l<p\). The main result of the paper is
Theorem 3.2. Assume that \(\gamma_1\gamma_2 \cdots\gamma_n=1\) for \(\gamma_1,\gamma_2,\ldots,\gamma_n\in G\) and a free order of \(\gamma_1\gamma_2\cdots\gamma_n\) (i.e., a product of the \(\gamma_i\) in an order obtained by some permutation) is equal \(g^q\) for a natural \(q\) which is not a multiple of \(p\). Then the rotation angle of \(g\) is admissible.


30F99 Riemann surfaces
Full Text: DOI


[1] E. Bujalance et al., On compact Riemann surfaces with dihedral groups of automorphisms, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, 465-477. · Zbl 1059.30030
[2] H. Glover and G. Mislin, Torsion in the mapping class group and its cohomology, J. Pure Appl. Algebra 44 (1987), no. 1-3, 177-189. · Zbl 0617.57005
[3] W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser. (2) 17 (1966), 86-97. · Zbl 0156.08901
[4] S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235-265. · Zbl 0528.57008
[5] K. Tsuboi, The finite group action and the equivariant determinant of elliptic operators, J. Math. Soc. Japan 57 (2005), no. 1, 95-113. · Zbl 1088.58016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.