Exceptional Discrete Mapping Class Group Orbits in Moduli Spaces

Abstract

Let $M$ be a four-holed sphere and $\Gamma$ the mapping class group of $M$ fixing $\partial M$. The group $\Gamma$ acts on the space ${\mathcal M}_{\mathcal B}(SU(2))$ of SU(2)-gauge equivalence classes of flat SU(2)-connections on $M$ with fixed holonomy on $\partial M$. We give examples of flat SU(2)-connections whose holonomy groups are dense in SU(2), but whose $\Gamma$-orbits are discrete in ${\mathcal M}_{\mathcal B}(SU(2))$. This phenomenon does not occur for surfaces with genus greater than zero.