Georgia Institute of Technology College of Sciences

Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2. Given a finite subgroup H < Mod(Sg), let Fix(H) denote the set of fixed points induced by the action of H on the Teichmuller space Teich(Sg). In this talk, we give an explicit description of Fix(H), when H is cyclic, thereby providing a complete solution to the Modular Nielsen Realization Problem for this case. Among other applications of these realizations, we derive an intriguing correlation between finite order maps and the filling systems of surfaces. Finally, we will briefly discuss some examples of realizations of two-generator finite abelian actions.