## Triangulations
and moduli spaces of Riemann surfaces with group actions

To appear in Manuscripta Mathematica

### Abstract

We study that subset of the moduli space
of stable
genus g, g>1, Riemann surfaces which consists of such
stable Riemann surfaces on
which a given finite group F acts. We show first that this subset is
compact. It turns out that, for general finite groups F, the above
subset is not connected. We show, however, that for Z_2
actions this subset * is* connected. Finally, we show that
even in the moduli space of * smooth* genus g Riemann surfaces,
the subset of those Riemann surfaces on which Z_2 acts * is*
connected. In view of deliberations of Klein, this was somewhat surprising.

These results are based on new coordinates for moduli spaces. These
coordinates are obtained by certain *regular* triangulations
of Riemann surfaces. These triangulations play an important
role also elsewhere, for instance
in approximating eigenfunctions of the Laplace operator numerically.

This work has been supported by the European Communities Science
Plan Project No SCI*-CT91 (TSTS) ``Computational Methods
in the Theory of Riemann Surfaces and Algebraic Curves,''
by Academy of Finland and by the Swiss National Science Foundation Grant
20-34099.92.

We thank M. C. Petrus for providing excellent motivation for this work.

September 17, 1995
M. Seppälä