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Planar representations of group actions on surfaces

The space of skeletal signatures was introduced as a simple but generally much coarser two-dimensional representation of the space of all signatures with which a group can act on a compact oriented surface. In this paper, we provide a complete characterization of the groups for which the skeletal signature space provides a complete and accurate picture of the structure of the full space of signatures. We show that such groups fall into three distinct families, and for two of these families, we show that for sufficiently large genus, the skeletal signature space depends essentially only on group order, and we explicitly describe these spaces. For the third family, we show that the skeletal signature space depends much more strongly upon group structure and provide some partial analysis of the structure of this space.