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

Tsuboi, Kenji

Proc. Japan Acad., Ser. A 84, No. 10, 184-185 (2008).

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.

Reviewer: Samyon R. Nasyrov (Kazan’)

MSC:

30F99

Riemann surfaces

Keywords:

Riemann surface; mapping class group; finite group; elliptic operator

References:

[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