## On $$pq$$-hyperelliptic Riemann surfaces.(English)Zbl 1080.30037

Let $$X$$ be a compact Riemann surface of genus $$g\geq 2$$. The surface $$X$$ is said to be $$p$$-hyperelliptic if it admits a conformal involution $$\rho$$, called a $$p$$-hyperelliptic involution, such that the quotient $$X/\rho$$ has genus $$p$$. Let us suppose that $$X$$ also admits a $$q$$-hyperelliptic involution. In such a case we say that $$X$$ is a $$pq$$-hyperelliptic surface. In this paper $$pq$$-hyperelliptic surfaces are studied and several bounds for the genus of the surface in terms of $$p$$ and $$q$$ are obtained.
It is proved that for arbitrary integers $$0\leq p\leq q$$ with $$(p,q)\neq (0,0)$$ the genus $$g$$ of a $$pq$$-hyperelliptic Riemann surface satisfies $$2q-1\leq g\leq 2p+2q+1$$. Conversely let us suppose that these numerical conditions are satisfied then there exists a Riemann surface admitting commuting $$p$$- and $$q$$-hyperelliptic involutions whose product is a $$t$$ -hyperelliptic involution if and only if $$t$$ is a nonnegative integer satisfying $$(g+1)/2-(p+1)\leq t\leq (g+1)/2$$ and such that $$p+q+t-g$$ is an even and nonnegative integer. Moreover if $$g\geq 2p+2q-2$$ then $$t=g-p-q$$. As a consequence there exist Riemann surfaces of genus $$2$$ or $$3$$ which are simultaneously hyperelliptic and elliptic-hyperelliptic ($$p=0$$, $$q=1$$) being the product of the $$0$$- and $$1$$-involutions a $$1$$- or a $$2$$-involution, respectively.
The author obtains several other results: if $$q>2p+1$$ then a $$p$$-hyperelliptic involution is unique and central in the full group of automorphisms of a $$pq$$ -hyperelliptic surface. It is proved that if $$(p,q)\neq (0,0)$$ and the genus $$g$$ satisfies $$3p+3q+2<2g\leq 4p+4q+2$$ then any $$p$$- and $$q$$-hyperelliptic involutions commute. Finally if $$p<q<2p$$ and $$3q+1<g\leq2p+2q+1$$ then $$p$$- and $$q$$-hyperelliptic involutions in a $$pq$$-hyperelliptic Riemann surface are unique and central in the full group of automorphisms. Hence if $$3q+2<g\leq4q+1$$, a Riemann surface of genus $$g$$ can admit at most two $$q$$-involutions.
Fuchsian signatures, the Hurwitz-Riemann formula and a result from A. M. Macbeath [Bull. Lond. Math. Soc. 5, 103–108 (1973; Zbl 0259.30016)] about the number of fixed points of an automorphism of a Riemann surface, are the main tools in the proofs, which are short and clear.

### MSC:

 30F10 Compact Riemann surfaces and uniformization 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)

Zbl 0259.30016
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