Topological classification of Zₚᵐ actions on surfaces

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Let S be a closed (compact without boundary) oriented surface with genus g, and G be a group isomorphic to % Zₚᵐ, where p is a prime integer. An action of G on S is a pair (S,f), where f is a representation of G in the group of orientation preserving autohomeomorphisms of S. Two actions (S,f) and (S′,f′) are called strongly (resp. weakly) equivalent if there is a homeomorphism, % ψ:S→ S′, sending the orientation of S to the orientation of S% ′, such that f′(h)=ψ∘ f(h)∘ ψ⁻¹, (resp. there is an automorphism α∈ Aut(G) such that f′∘ α(h)=ψ∘ f(h)∘ ψ⁻¹) for all h∈ G. We give the full description of strong and weak equivalence classes. The main idea of our work is the fact that a fixed point free action of Zₚᵐ on a surface provides a bilinear antisymmetric form on Zₚᵐ. For instance, we prove that the weakly equivalence classes of actions of G on surfaces with orbit space of genus g are in one to one correspondence with the set of pairs which consist in a positive integer number k, k≤ m-n, k=(m-n)% mod2, g≥ 1/2(m-n+k), and an orbit of the action of % Aut(G) on the set of unordered r-tuples [C₁,...,Cᵣ] of non-trivial elements generating a subgroup isomorphic to Zₚⁿ and such that ∑₁ʳCᵢ=0. We use this result in describing the moduli space of complex algebraic curves admitting a group of automorphisms isomorphic to Zₚᵐ.