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We determine all finite maximal elementary abelian group actions on compactoriented surfaces of genus $\sigma\geq 2$ which are unique up to topologicalequivalence. For certain special classes of such actions, we determine groupextensions which also define unique actions. In addition, we explore in detailone of the families of such surfaces considered as compact Riemann surfaces andtackle the classical problem of constructing defining equations.

Topologically unique maximal elementary Abelian group actions on compact oriented surfaces