Is the Mapping Class Group Always the Biggest Group?
The mapping class group of a surface is a very important, but still mysterious group. Natural actions of the mapping class group appear on representation varieties of surface groups. In some cases, e.g. when this action preserves a metric, we know that no bigger group can act by isometries. But what happens if we look at actions that preserve not a metric, but only the natural symplectic structure on representation varieties?