On fixed points on compact Riemann surfaces
Bull. Korean Math. Soc. 2011 Vol. 48, No. 5, 1015-1021
Printed September 1, 2011
Grzegorz Gromadzki
University of Gda\'nsk
Abstract : A point of a Riemann surface $X$ is said to be its fixed point if it is a fixed point of one of its nontrivial holomorphic automorphisms. We start this note by proving that the set ${\rm Fix}(X)$ of fixed points of a Riemann surface $X$ of genus $g\geq 2$ has at most $82(g-1)$ elements and this bound is attained just for $X$ having a Hurwitz group of automorphisms, i.e., a group of order $84(g-1)$. The set of such points is invariant under the group of holomorphic automorphisms of $X$ and we study the corresponding symmetric representation. We show that its algebraic type is an essential invariant of the topological type of the holomorphic action and we study its kernel, to find in particular some sufficient conditions for its faithfulness.
Keywords : automorphisms of Riemann surfaces, fixed point, Fuchsian groups, symmetric representation
MSC numbers : Primary 30F10; Secondary 14H37, 14H55, 20B25
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