Bounds for the order of supersoluble automorphism groups of Riemann surfaces

Abstract:

The maximal automorphism groups of compact Riemann surfaces for a class of groups positioned between nilpotent and soluble groups is investigated. It is proved that if $G$ is any finite supersoluble group acting as the automorphism group of some compact Riemann surface $\Omega$ of genus $g \geq 2$, then: (i) If $g = 2$ then $|G| \leq 24$ and equality occurs when $G$ is the supersoluble group ${D_4} \otimes {{\mathbf {Z}}_3}$ that is the semidirect product of the dihedral group of order 8 and the cyclic group of order 3. This exceptional case occurs when the Fuchsian group $\Gamma$ has the signature (0;2,4,6), and can cover only this finite supersoluble group of order 24. (ii) If $g \geq 3$ then $|G| \leq 18\left ( {g - 1} \right )$, and if $|G| = 18\left ( {g - 1} \right )$ then $\left ( {g - 1} \right )$ must be a power of 3. Conversely if $\left ( {g - 1} \right ) = {3^n},n \geq 2$, then there is at least one surface $\Omega$ of genus $g$ with an automorphism group of order $18\left ( {g - 1} \right )$ which must be supersoluble since its order is of the form $2{3^m}$. This bound corresponds to a specific Fuchsian group given by the signature (0;2,3,18). The terms in the chief series of each of these Fuchsian groups to the point where a torsion-free subgroup is reached are computed.