Classification of ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} actions on surfaces

Antonio F. Costa  Sergey M. Natanzon

Abstract. In this article we classify the fixed point free actions of the groups ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} (where p is a prime integer) on compact oriented surfaces.

1 Introduction

Finite abelian group actions on surfaces by autohomeomorphisms constitutes a classical subject, see the articles [N], [E], [J1], [J2], [Na1], [S], [Z], [CN]. In our article we present a direct and complete way to deal with the topological classification of fixed point free ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} actions, where p is a prime integer (ℤ_{p^{s}}^{m} = ℤ_{p^{s}}⊕ ...^{m} ⊕ ℤ_{p^{s}} and ℤ_{p^{s}} = ℤ/p^{s}ℤ). Our results show some incorrections in [J1] (compare, for example, Proposition 4.6 in [J1] with our Corollary 15).

An important part of our work is the classification of skew symmetric forms on ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} (Theorem 11), since a fixed point free action of ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} on a surface provides an alternating or skew symmetric bilinear form on ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}. We define a signature to classify such skew symmetric forms (Definition 12) and that signature classify the actions on surfaces of a given genus (Theorem 13), considering the weak equivalence of actions (in the terminology of Edmonds [E]). We deal with the strong equivalence in Theorem 8 and the realization of the possible actions is in Theorem 9.

Since G is a finite group, given an action (S̃, f) of G, it is possible to construct an analytic structure on S such that f(G) is a group of automorphisms as Riemann surface (see [K]). Hence all the actions considered in this paper appear as automorphism group actions of complex algebraic curves. One of the motivations for our study is the description of the set of connected components in the moduli space M of pairs (C, G), where C is a complex algebraic curve and G ≅ ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} is a group of fixed point free automorphisms of C. According to [Na2] the description of connected components of M is reduced to the description of topological classes of pairs (S̃, K) where K is a group of autohomeomorphisms of S̃ and K is isomorphic to ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}. We consider that (S̃, K) and (S̃′, K′) are equivalent if there exist a homeomorphism φ : S̃ → S̃′ such that K′ = φ ∘ K ∘ φ^{−1}. These equivalence classes are in one to one correspondence with classes of weak equivalence.

0Research partially supported by grants RFBR-01-01-00739, NSh-1972.2003.1 and Ministerio de Ciencia y Tecnología BFM2002-04801.

Acknowledgement. This paper was written during the authors stay at Max-Planck-Institut für Mathematik in Bonn. We would like to thank this institution for its support and hospitality.

2 Algebraic preliminaries.

Let us consider the standard lattice ℤ^{2g} = ℤ⊕ ...^{2g} ⊕ ℤ with the standard basis (e_{i}) = ((0, ..., 1^{(i)}, ...0)). We define the skew symmetric bilinear form (., .) : ℤ^{2g} × ℤ^{2g} → ℤ, by (e_{i}, e_{j}) = δ_{i+j,2g+1} for i < j.

We consider also the group ℤ_{p^{2}}^{2g} = ℤ_{p^{2}}⊕ ...^{2g} ⊕ ℤ_{p^{2}} where p is a prime and ℤ_{p^{2}} = ℤ/p^{2}ℤ = \{\overline{0}, \overline{1}, \overline{2}, ..., \overline{p^{2} − 1}\}. Let φ : ℤ^{2g} → ℤ_{p^{2}}^{2g} be the natu-

ral projection defined by φ(e_{i}) = \overline{e_{i}}, where \overline{e_{i}} = (0, ..., \overline{1}^{(i)}, ...0). Then we have a skew symmetric bilinear form (., .)_{p^{2}} : ℤ_{p^{2}}^{2g} × ℤ_{p^{2}}^{2g} → ℤ_{p^{2}} defined by (\overline{e_{i}}, \overline{e_{j}}) = (φ(e_{i}), φ(e_{j}))_{p^{2}} = (e_{i}, e_{j}) mod p^{2}. We shall denote by ℤ̂_{p^{2}}^{2g} the group ℤ_{p^{2}}^{2g} provided with the bilinear form (., .)_{p^{2}}.

Let Sp(2g, ℤ) and Sp_{p^{2}}(2g, ℤ_{p^{2}}) be the subgroups of the automorphisms groups of ℤ^{2g} and ℤ_{p^{2}}^{2g} that preserve the bilinear forms (., .) and (., .)_{p^{2}} respectively. The natural projection φ : ℤ^{2g} → ℤ_{p^{2}}^{2g} induces a homomorphism φ_{∗} : Sp(2g, ℤ) → Sp_{p^{2}}(2g, ℤ_{p^{2}}) such that φ_{∗}(f)∘φ = φ∘f for all f ∈ Sp(2g, ℤ).

The following result is known:

Lemma 1 (See Lemma 4.1, pg 177 of [E]) φ_{∗}(Sp(2g, ℤ)) = Sp_{p^{2}}(2g, ℤ_{p^{2}}).

We shall also need the following result:

Theorem 2 (Witt’s Theorem, see Theorem 3.1, pg. 175 of [E]) Let G, G′ be subgroups of ℤ̂_{p^{2}}^{2g} both subgroups isomorphic to ℤ_{p^{2}}^{r} and ψ : G → G′ be an isomorphism such that (ψ(a), ψ(b))_{p^{2}} = (a, b)_{p^{2}} for all a, b ∈ G. Then there is an automorphism ψ̃ ∈ Sp_{p^{2}}(2g, ℤ_{p^{2}}) such that ψ̃ restricted to G is ψ.

Corollary 3 Let G, G′ be subgroups of ℤ̂_{p^{2}}^{2g} and ψ : G → G′ be an isomorphism such that (ψ(a), ψ(b))_{p^{2}} = (a, b)_{p^{2}} for all a, b ∈ G. Then there is an automorphism ψ̃ ∈ Sp_{p^{2}}(2g, ℤ_{p^{2}}) such that ψ̃ restricted to G is ψ.

3 Strong classification of fixed point free orientation preserving actions of ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} on surfaces

Let G be a group isomorphic to ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}, where p is a prime integer. An (orientation preserving) action of G on an orientable surface of genus g̃ is a pair (S̃, f), where S̃ is a closed (compact without boundary) oriented surface with genus g̃, and f is a exact representation of G in the group of orientation preserving autohomeomorphisms of S̃. We shall consider fixed point free actions.

Definition 4 (Strong equivalence). Two actions (S̃, f) and (S̃′, f′) are called strongly equivalent if there is a homeomorphism, ψ̃ : S̃ → S̃′, sending the orientation of S̃ to the orientation of S̃′ and such that f′(h) = ψ̃ ∘ f(h) ∘ ψ̃^{−1}, for all h ∈ G.

We are interested in finding all strong equivalence classes of actions of ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}

We denote by S = S̃/f(G) and by φ = φ(f) : S̃ → S the natural projection. We shall consider the case when f(h) has no fixed points for any h ∈ G − {id}, i. e. the action of (S̃, f) is fixed point free. Then the projection φ(f) : S̃ → S is an unbranched covering with deck transformation group f(G).

Let us consider π_{1}(S) as the group of deck transformations of the universal covering of S. Then we have:

ω(S̃, f) : π_{1}(S) → π_{1}(S)/π_{1}(S̃) = f(G) →^{f^{−1}} G.

The resulting epimorphism ω(S̃, f) : π_{1}(S) → G is the monodromy epimorphism of the covering φ(f) : S̃ → S. The epimorphism ω(S̃, f) : π_{1}(S) → G, induces the epimorphism θ_{p^{2}}(S̃, f) : H_{1}(S, ℤ_{p^{2}}) → G, since G is abelian.

Conversely, given an epimorphism θ_{p^{2}} : H_{1}(S, ℤ_{p^{2}}) → G, there is an action (S̃, f) such that θ_{p^{2}} = θ_{p^{2}}(S̃, f). To obtain S̃ it is enough to consider the

monodromy ω : π_{1}(S) → H_{1}(S, ℤ_{p^{2}}) →^{θ_{p^{2}}} G and then S̃ = U/ ker ω, where U is the universal covering of S and the action of G is given by G = π_{1}(S)/ ker ω.

Definition 5 Let S and S′ be two surfaces. Two epimorphisms θ : H_{1}(S, ℤ_{p^{2}}) → G and θ′ : H_{1}(S′, ℤ_{p^{2}}) → G are called strongly equivalent if there is an orientation preserving homeomorphism ψ : S → S′ inducing an isomorphism ψ_{p^{2}} : H_{1}(S, ℤ_{p^{2}}) → H_{1}(S′, ℤ_{p^{2}}) such that θ = θ′ ∘ ψ_{p^{2}}.

Using covering space theory we have that:

Theorem 6 Two actions (S̃, f) and (S̃′, f′) are strongly equivalent if and only if the epimorphisms θ_{p^{2}}(S̃, f) and θ_{p^{2}}(S̃′, f′) are strongly equivalent.

Definition 7 Let (S̃, f) be an action of G, S = S̃/f(G), and θ = θ_{p^{2}}(S̃, f) : H_{1}(S, ℤ_{p^{2}}) → G be the epimorphism defined by the action (S̃, f). Let us consider the spaces of homomorphisms G^{∗} = {e : G → ℤ_{p^{2}}} and H^{1}(S, ℤ_{p^{2}}) = {e : H_{1}(S, ℤ_{p^{2}}) → ℤ_{p^{2}}}. Then θ generates a monomorphism θ^{∗} = θ^{∗}(S̃, f) : G^{∗} → H^{1}(S, ℤ_{p^{2}}). The intersection form (., .)_{p^{2}} = (., .)_{p^{2}}^{S} on H_{1}(S, ℤ_{p^{2}}) induces an isomorphism i : H^{1}(S, ℤ_{p^{2}}) → H_{1}(S, ℤ_{p^{2}}) defined by (a, .) → a and a form (., .)_{(S̃,f)} : G^{∗} × G^{∗} → ℤ_{p^{2}} such that (a, b)_{(S̃,f)} = (i ∘ θ^{∗}(a), i ∘ θ^{∗}(b))_{p^{2}}.

Theorem 8 Two actions (S̃, f) and (S̃′, f′) of the group G are strongly equivalent if and only if S̃ and S̃′ have the same genus and (., .)_{(S̃,f)} = (., .)_{(S̃′,f′)}.

The proof is a direct adaptation to our case of the proof of Theorem 8 in [CN]. It is necessary to use the Corollary 3 and the Lemma 1 in the above Section instead of the Witt’s Theorem in [A, 3.9] and the Theorems VII.20, VII.21 of [Ne] (as used in [CN]).

Theorem 9 Let G ≅ ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}, (., .) : G × G → ℤ_{p^{2}} be an alternating bilinear form. Then there exists an action (S̃, f) such that (., .) = (., .)_{(S̃,f)} and the genus of S̃/f(G) is g if and only if G ↪ ℤ̂_{p^{2}}^{2g}.

Proof. First we construct the action from the form and the numerical conditions. We have a basis of G, (x_{i}, (i = 1, ..., m)). Now we consider a surface S of genus g and applying the fact that Witt problems are solvable in ℤ_{p^{2}}^{2g}, (see [E, pg. 175]) we construct a basis of H_{1}(S, ℤ_{p^{2}}), (χ_{i}, (i = 1, ..., m), ν_{i}, (i = m + 1, ..., 2g)), such that (χ_{i}, χ_{j}) = (x_{i}, x_{j}). Then we construct the epimorphism θ : H_{1}(S, ℤ_{p^{2}}) → G, defined by θ(χ_{i}) = x_{i}, if i ≤ m, θ(ν_{i}) = 0. Then the epimorphism θ defines a homomorphism H_{1}(S, ℤ) →H_{1}(S, ℤ_{p^{2}}) → G and thus a regular covering S̃ → S with automorphism group G. The action of G on S̃ satisfies (., .)_{(S̃,f)} = (., .).

Conversely if there is an action (S̃, f) such that (., .) = (., .)_{(S̃,f)} it is obvious, by the construction of (., .)_{(S̃,f)}, that G ↪ ℤ_{p^{2}}^{2g}.

4 Weak classification of fixed point free orientation preserving actions of ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} on surfaces

Definition 10 (Weak equivalence) Let (S̃, f) and (S̃′, f′) be two actions of a group G. We shall say that (S̃, f) and (S̃′, f′) are weakly equivalent if there is a homeomorphism ψ̃ : S̃ → S̃′ and an automorphism α ∈ Aut(G) such that f′ ∘ α(h) = ψ̃ ∘ f(h) ∘ ψ̃^{−1}, h ∈ G.

We need an algebraic theorem of classification of skew-symmetric bilinear forms on ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}.

Theorem 11 Let V be ℤ_{p2}^{m} ⊕ ℤ_{p}^{n} and ξ : V × V → ℤ_{p2} be a skew-symmetric bilinear form. Then there exists a factorization V = V_{2,2|2} ⊕ V_{2,2|1} ⊕ V_{2,2|0} ⊕ V_{2,1|1} ⊕ V_{1,1|1} ⊕ V_{1,1|0} such that V_{s,s|t} ≅ ℤ_{p^{s}}^{v_{s.s|t}}, V_{2,1|1} ≅ (ℤ_{p2} ⊕ ℤ_{p})^{v_{∗}}, v_{∗} ≤ 1, Ker(ξ) = V_{2,2|0} ⊕ V_{1,1|0}, ξ(V_{s_{a},s_{b}|t}, V_{k_{a},k_{b}|l}) = 0 for (s_{a}, s_{b}|t) ≠ (k_{a}, k_{b}|l) and v_{2,2|0} = 0 if v_{1,1|1} > 0.

Moreover we can construct the factorization such that the restriction ξ on V_{s_{a},s_{b}|2} and V_{s_{a},s_{b}|1} is nondegenerated, and V_{s,s|2}, V_{s,s|1} have bases {a_{s|t}^{i}, b_{s|t}^{i}|i = 1, ..., k_{s|t}} such that ξ(a_{s|t}^{i}, b_{s|t}^{j}) = p^{2−t}δ_{ij}.

The dimensions of V_{s_{a},s_{b}|t} are uniquely defined by ξ and thus form a complete system of invariants for ξ.

Proof. For s, t ∈ {1, 2} call a set {a^{i}, b^{i}|i = 1, ..., k} (s|t)-symplectic system, if it consist of elements of order p^{s} and ξ(a^{i}, b^{j}) = p^{(2−t)}δ_{ij}.

Let B_{2|2} = {a_{2|2}^{i}, b_{2|2}^{i}|i = 1, ..., k_{2|2}} be a (2|2)-symplectic system in V with maximal cardinality. Let V_{2,2|2} be the submodule generated by B_{2|2}. We define W_{2,2|2} = {v ∈ V |ξ(v, V_{2,2|2}) = 0}. Then V = V_{2,2|2} ⊕ W_{2,2|2} and moreover pξ(v_{1}, v_{2}) = 0 for v_{1}, v_{2} ∈ W_{2,2|2}.

Let B_{2|1} = {a_{2|1}^{i}, b_{2|1}^{i}|i = 1, ..., k_{2|1}} be a (2|1)-symplectic system in W_{2,2|2} with maximal cardinality. Let V_{2,2|1} be the submodule generated by B_{2|1}. Now we define W_{2,2|1} = {v ∈ W_{2,2|2}|ξ(v, V_{2,2|1}) = 0}. Then W_{2,2|2} = V_{2,2|1} ⊕ W_{2,2|1} and moreover ξ(v_{1}, v_{2}) = 0 if v_{1}, v_{2} ∈ W_{2,2|1} are elements of order p^{2}.

We define W_{2,1|1} = W_{2,2|1} if W_{2,2|1} − W_{2,2|1} ∩ Ker(ξ) does not contain elements of order p^{2}. In opposite case there exist an element a of order p^{2} and an element b of order p such that ξ(a, b) = p. In this case W_{2,1|1} = {v ∈ W_{2,2|1}|ξ(v, a) = ξ(v, b) = 0}. Note that there is no possible the existence of a_{1}, a_{2} of order p^{2} and b_{1} of order p such that ξ(a_{1}, a_{2}) = 0, ξ(a_{i}, b_{1}) = pδ_{i1}. In such a case we can add {a_{1}, a_{2}+b_{1}} to the system B_{2|1} and this fact contradicts the maximality of B_{2|1}.

Let B_{1|1} = {a_{1|1}^{i}, b_{1|1}^{i}|i = 1, ..., k_{2|1}} be a (1|1)-symplectic system in W_{2,1|1} with maximal cardinality. Let V_{1,1|1} be the submodule generated by B_{1|1}. We define W_{1,1|1} = {v ∈ W_{2,1|1}|ξ(v, V_{1,1|1}) = 0}. Then W_{2,1|1} = V_{1,1|1} ⊕ W_{1,1|1} and W_{1,1|1} = Ker(ξ) ≅ V_{2,2|0} ⊕ V_{1,1|0} where V_{s,s|0} ≅ ℤ_{p^{s}}^{v_{s|0}}.

Now we shall prove that v_{2,2|0} = 0 if v_{1,1|1} > 0. Assume that v_{1,1|1} > 0. If W (2, 2|1) contain two linear independent elements v_{1}, v_{2} of order p^{2} then either ξ(v_{1} +a_{1,1|1}^{1}, v_{2} +b_{1,1|1}^{1}) ≠ 0 or ξ(v_{1} +a_{1,1|1}^{1}, v_{2}) ≠ 0 or ξ(v_{1}, v_{2} +b_{1,1|1}^{1}) ≠ 0 contradicting the maximal cardinality of B_{2|1}. Assume that W (2, 2|1) contains an element v of order p^{2}, hence either ξ(v, b_{1,1|1}^{1}) ≠ 0 or ξ(v + a_{1,1|1}^{1}, b_{1,1|1}^{1}) ≠ 0. Thus v_{∗} = 1 and v_{2,2|0} = 0.

The dimensions of V_{s_{a},s_{b}|t} are defined by n, m, n′, m′ and m” where Ker(ξ) ≅ ℤ_{p2}^{m′} ⊕ ℤ_{p}^{n′}, Ker(pξ) ≅ ℤ_{p2}^{m”} ⊕ ℤ_{p}^{n}. From the factorization follows that m = v_{2,2|2}+v_{2,2|1}+v_{2,2|0}+v_{∗}, m′ = v_{2,2|0}, m” = v_{2,2|0}+v_{2,2|1}+v_{∗}, n = v_{1,1|1}+v_{1,1|0}+v_{∗}, n′ = v_{1,1|0}. Moreover v_{∗} = 0 if m − m′ is even and v_{∗} = 1 if m − m′ odd.

Definition 12 Let V be ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} and ξ : V × V → ℤ_{p^{2}} be a skew-symmetric bilinear form. Let V = V_{2,2|2} ⊕ V_{2,2|1} ⊕ V_{2,2|0} ⊕ V_{2,1|1} ⊕ V_{1,1|1} ⊕ V_{1,1|0} be the factorization of V produced by ξ and given in the above Theorem. We shall call signature of ξ to the ordered set (v_{2,2|2}, v_{2,2|1}v_{2,2|0}, v_{∗}, v_{1,1|1}, v_{1,1|0})

The next Theorem solves the problem of weak classification of fixed point free actions of ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} on surfaces:

Theorem 13 Let (S̃, f) and (S̃′, f′) be two actions of a group G ≅ ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n}. Let (., .)_{(S̃,f)} and (., .)_{(S̃′,f′)} be the skew symmetric bilinear forms induced by the two actions. Then the actions (S̃, f) and (S̃′, f′) are weakly equivalent if and only if the forms (., .)_{(S̃,f)} and (., .)_{(S̃′,f′)} have the same signature and S̃, S̃′ have the same genus.

Proof. Let us call S = S̃/f(G) and S′ = S̃′/f′(G). Let θ^{∗}(S̃, f) and θ^{∗}(S̃′, f′) be the epimorphisms defined by the two actions, G̃ be the image of G^{∗} in H_{1}(S, ℤ_{p^{2}}) by θ^{∗}(S̃, f) and G̃′ be the image of G^{∗} in H_{1}(S′, ℤ_{p^{2}}) by θ^{∗}(S̃′, f′).

Let (., .)_{(S̃,f)} and (., .)_{(S̃′,f′)} be the skew symmetric bilinear forms induced by the two actions. Assume that (., .)_{(S̃,f)} and (., .)_{(S̃′,f′)} have the same signature then there exists an isomorphism ψ : G̃′ → G̃ such that (ψ(a), ψ(b))_{(S̃′,f′)} = (a, b)_{(S̃,f)}.

Assume now that S̃, S̃′ have the same genus, then g(S) = g(S′) (the actions are fixed point free). Using Corollary 3, there is an isomorphism ψ̃ :

H^{1}(S′, ℤ_{p^{2}}) → H^{1}(S, ℤ_{p^{2}}) giving by restriction ψ and sending the intersection form of H^{1}(S′, ℤ_{p^{2}}) to the intersection form of H^{1}(S, ℤ_{p^{2}}). By Lemma 1 and [MKS, page 178], there exists a homeomorphism φ : S → S′ inducing ψ̃ on cohomology. Then by Theorem 8, the actions (S̃, f) and (S̃, φ^{−1} ∘ f′ ∘ φ) are strongly equivalent. The isomorphism ψ, defines an automorphism of G giving the weak equivalence between (S̃, f) and (S̃′, f′).

Note that, if ℤ_{p^{2}}^{m} ⊕ ℤ_{p}^{n} acts on a surface of genus g̃, then the genus of the quotient surface is g = \frac{g̃−1}{p^{2m+n}} + 1. The following corollary is one of the main results of this paper:

Corollary 14 Weak equivalence classes of fixed point free ℤ_{p^{2}}^{m} ⊕ℤ_{p}^{n} actions on surfaces of genus g̃ are in bijection with the signatures of subgroups of ℤ̂_{p^{2}}^{2g}.

Proof. Theorem 13 tells us that each signature determines a weak equivalence class. By Theorem 9 each signature of a subgroup of ℤ̂_{p^{2}}^{2g} defines a non-empty weak equivalence class.

Corollary 15 Weak equivalence classes of fixed point free ℤ_{p^{2}}^{m} ⊕ℤ_{p}^{n} actions on surfaces of genus g̃ are in bijection with the ordered set of integers (v_{2,2|2}, v_{2,2|1}, v_{2,2|0}, v_{∗}, v_{1,1|1}, v_{1,1|0}) such that v_{2,2|2}, v_{2,2|1}, v_{1,1|1} are even, v_{∗} < 2, v_{2,2|0} = 0 if v_{1,1|1} > 0 and m = v_{2,2|2} + v_{2,2|1} + v_{2,2|0} + v_{∗}, n = v_{1,1|1} + v_{1,1|0} + v_{∗}.

Example. We shall construct an action of ℤ_{4}^{5} ⊕ ℤ_{2}^{4} on a surface of genus 2^{16} + 1 (with quotient surface of genus 5) and inducing a bilinear form with signature (2, 2, 0, 1, 2, 1). Let e_{1}, e_{2}, e_{3}, e_{4}, e_{5} be a base of ℤ_{4}^{5} and f_{1}, f_{2}, f_{3}, f_{4} be a base of ⊕ℤ_{2}^{4} Let S be a surface of genus 5 and (a_{i}, b_{i}; i = 1, ..., 5) be a symplectic base of H^{1}(S, ℤ_{4}). The action is defined by θ : H^{1}(S, ℤ_{4}) → ℤ_{4}^{5} ⊕ℤ_{2}^{4}, where θ(a_{1}) = e_{1}, θ(b_{1}) = e_{2}, θ(a_{2}) = e_{3}, θ(b_{2}) = e_{4}, θ(a_{3}) = e_{5}, θ(b_{3}) = f_{1}, θ(a_{4}) = f_{2}, θ(b_{4}) = f_{3}, θ(a_{5}) = f_{4}, θ(b_{5}) = 0

The above example shows also the way of geometrical realization of a representation of each weak equivalence class of action. This gives a proof of Corollary 15.

References

[A] E. Artin, Geometric Algebra, Interscience, New York, 1957.

[CN] A. F. Costa and S. M. Natanzon, Topological Classification of ℤ_{p}^{m} Actions on Surfaces, Michigan Math. J . 50 (2002) 451-460.

[E] A. L. Edmonds, Surface symmetry I, Michigan Math. J., 29, (1982), 171-183.

[J1] S. A. Jassim, Finite abelian coverings, Glasgow Math. J., 25, (1984), 207-218.

[J2] S. A. Jassim, Finite abelian actions on surfaces, Glasgow Math. J., 35, (1993), 225-234.

[K] B. Kerekjarto, Vorlesungen über Topologie. I. Flächentopologie. Springer-Verlag, Berlin, 1923.

[MKS] W. Magnus, A. Karras and D. Solitar, Combinatorial group theory, Dover, New York, 1976.

[Na1] S. M. Natanzon, Moduli spaces of complex algebraic curves with isomorphic to (ℤ/2ℤ)^{m} groups of automorphisms, Differential Geometry and its Applications 5, (1995) 1-11.

[Na2] S. M. Natanzon, Moduli of Riemann surfaces, Hurwitz-type spaces, and their superanalogues, Russian Math. Surveys, 54:1 (1999) 61-117.

[Ne] M. Newman, Integral Matrices, Academic Press, New York, 1972.

[N] J. Nielsen, Die Struktur periodischer Transformationen von Flächen, Danske Vid. Selsk. Mat.-Fys. Medd. 15 (1937), 1-77.

[S] P. A. Smith, Abelian actions of periodic maps on compact surfaces, Michigan Math. J. 14 (1967), 257-275.

[Z] B. Zimmermann, Finite abelian group actions on surfaces, Yocohama Mathematical Journal, 38, (1990) 13-21.

Antonio F. Costa Departamento de Matemáticas Fundamentales,UNED 28040-Madrid, Spain. e-mail: acosta@mat.uned.es

Sergey M. Natanzon Moscow State University and Independent University of Moscow, Moscow, Russia. e-mail: natanzon@mccme.ru