A well-known theorem of Hurwitz states that if τ[ratio ]S→S
is a conformal self-mapping of a compact Riemann surface of genus
g[ges ]2, then it has at most 2g+2
fixed points and that equality occurs if and only if τ is a hyperelliptic
involution.
In this paper we consider this problem for a K-quasiconformal
self-mapping
f[ratio ]S→S. The result we obtain is that
the
number of fixed points (suitably counted) is bounded by
2+g(K1/2+K−1/2),
and that this bound is sharp. We see that when
K=1, that is, when f is conformal, our result agrees
with the classical one.