## 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.

### MSC:

 30F99 Riemann surfaces
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### References:

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