Volume 12, issue 1 (2012)

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A lower bound for the number of group actions on a compact Riemann surface

James W Anderson and Aaron Wootton

Algebraic & Geometric Topology 12 (2012) 19–35
Abstract

We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus $\sigma \ge 2$ is at least quadratic in $\sigma$. We do this through the introduction of a coarse signature space, the space ${\mathsc{K}}_{\sigma }$ of skeletal signatures of group actions on compact Riemann surfaces of genus $\sigma$. We discuss the basic properties of ${\mathsc{K}}_{\sigma }$ and present a full conjectural description.

Keywords
Riemann surface, automorphism, signature, mapping class group
Mathematical Subject Classification 2010
Primary: 14H37
Secondary: 57M60, 30F20