On topological type of periodic self-homeomorphisms of closed non-orientable surfaces

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Introduction
By an effective action of a finite group G on a closed surface S we understand an embedding of G into the group Homeo(S) of homeomorphisms of S. Two such actions are topologically equivalent, or of the same topological type, if the images of G are conjugate in Homeo(S). The topological classification of finite group actions on closed surfaces is a classical problem going back to Nielsen [16].
Let M g denote the moduli space of complex algebraic curves of genus g ≥ 2 and consider its subset M g (G) consisting of points representing curves with a finite group G of birational automorphisms. It is intuitively plausible, and Teichmüller-Royden theory provides a more precise justification, that M g (G) is smaller for bigger G. In other words, a curve is better described by its group of automorphims when this group is bigger. By the famous Hurwitz bound its order does not exceed 84(g − 1).
Particularly interesting are the cases when a curve X is determined, up to birational equivalence, by the topological type of the action of G, or only by its ramification data, by which we understand the genus of the orbit space X/G and the branching indices of the projection X → X/G, or even only by the order of G. More specifically, when |G| > 12(g−1), then by the Hurwitz-Riemann formula and elementary Teichmüller theory, M g (G) is finite and topological and birational types of the action coincide, see [9,12,17,18]. For example, the main discovery of Nakagawa from [15], independently proved also by Hirose in [13] using less explicit methods which are closer to our appproach, asserts that with a few exceptions, a complex curve of genus g ≥ 2 having an automorphism of order N ≥ 3g is determined (up to birational equivalence) by N . The condition N ≥ 3g turns out to be quite restrictive, as it forces N to be one of 4g + 2, 4g, 3g + 3 or 3g for g > 12. On the other hand, there are infinitely many rational numbers q and r , such that for infinitely many g ≥ 2 there is a homeomorphism of a closed orientable surface of genus g having order N = qg + r satisfying 2(g − 1) < N < 3g. In [1] a more general situation is studied, when the order of a cyclic group of automorphisms of a compact Riemann surface of genus g ≥ 2, or the ramification data of the action, determine its topological type. Importance of such results follows from their connection with topology of the singular locus of the moduli space of complex algebraic curves, see [11].
Motivated by [1,13], in this paper we consider analogous problems for purely imaginary real curves, which can also be seen as compact, unbordered, non-orientable surfaces with dianalitic structure (see [5] for a definition). The study of dianalitic automorphisms of such surfaces is equivalent to the study of their periodic self-homeomorphisms, because every periodic homeomorphism of a surface S g of topological genus g ≥ 3 is a dianalitic automorphism with respect to some dianalitic structure on S g . Bujalance showed in [2] (see also later paper of S. Wang [19]) that the maximal order of such automorphism of a non-orientable surface of genus g ≥ 3 is equal to 2g or 2(g − 1) for g odd or even respectively, and this upper bound is attained for all g ≥ 3. The case g = 3 is well understood, as the mapping class group of S 3 is isomorphic to GL 2 (Z) (see [8]), and the classification of conjugacy classes of torsion elements in the latter group is known. Another interesting problem concerning cyclic periodic actions on non-orientable surfaces was considered in recent paper [3], where the authors investigated such actions which can not be extended to any bigger group.
Throughout the whole paper we denote by S or S g a closed (i.e. compact and without boundary) non-orientable surface of genus g ≥ 3. In this paper we study the extent to which the order N or the ramification data of a cyclic group G acting on S g determine the topological type of the action, which is important in virtue of the connection with topological properties of moduli spaces of purely imaginary real algebraic curves, similar as in the case of orientable surfaces. More specifically, in Sect. 3 we investigate rigidity of topological type of cyclic group actions of order N > g − 2 with prescribed ramification data. We consider a quite large family of actions, where the order N has the form N = qg + r , for infinitely many rational q and r . Furthermore, for each such pair q, r , there is an action of Z N on S g for infinitely many genera g. As an application, in Sect. 4 we calculate, for a fixed g and all N between max{g, 3 2 (g − 2)} and 2g, the numbers of topological types of action on S g of a cyclic group of order N . This should be seen as an analogue, for a non-orientable surface, of the main result of [13].

Principal definitions
Our approach is based on algebraic properties of discrete subgroups of isometries of the hyperbolic plane H, called NEC-groups. We refer the reader to the monograph [5] for an extensive exposition of the theory. Suppose that a finite group G acts effectively by homeomorphisms on a closed nonorientable surface S = S g of genus g ≥ 3. Fix a dinanalitic structure on S, with respect to which G acts by dianalitic automorphisms. Then S is conformally isomorphic to the orbit space H/ for a torsion-free NEC group isomorphic to π 1 (S). Such is called nonorientable surface group. Furthermore, G is isomorphic to the quotient / , for some other NEC-group , a subgroup of the normalizer of in the group of all isometries of H ( is equal to that normaliser if and only if G is the full group of dianalitic automorphisms of S). Equivalently, there is an epimorphism θ : → G with kernel , usually called smooth epimorphism to underline the fact that its kernel is torsion-free. This motivates the following definition.

Definition 2.1
Suppose that is an NEC group, G is a finite group, and θ : → G is an epimorphism. We say that θ is an NSK-map (non-orientable-surface-kernel-map) if and only if ker θ is a non-orientable surface group.
Two effective actions of G on S g are topologically conjugate (by a homeomorphism of S g ) if and only if the associated NSK-maps are equivalent in the sense of the next definition (see [4,Proposition 2.2] and its proof; the same argument applies to closed surfaces).

Definition 2.2
We say that two NSK-maps θ i : i → G, i = 1, 2, are equivalent if and only if there exist isomorphisms φ : 1 → 2 and α : G → G such that the following diagram is commutative.
The ramification data of G is encoded in the signature σ ( ) of , which in our case has the form where k > 0 if the sign is "+" (see [2]). The orbit space S/G = H/ has genus h and k boundary components, and it is orientable if and only if the sign is "+". From the signature one can also read a presentation of in terms of canonical generators and defining relations as follows. The generators are: The defining relations are: Note that is a proper NEC group, i.e. it contains orientation-reversing isometries. Among the canonical generators, c j and d l are orientation-reversing, the remaining ones are orientation-preserving. We denote by + the canonical Fuchsian subgroup of , consisting of all orientation-preserving elements of . Finally, we have the Hurwitz-Riemann ramification formula is the normalized hyperbolic area of a (arbitrary) fundamental region for . Here α = 2 if the sign of the signature is "+" and α = 1 otherwise.
The following lemma, which is a special case of [2, Proposition 3.2], provides an effective criterion for an NSK-map.

Lemma 2.3 Suppose that is an NEC group with signature (2.2). A group homomorphism θ : → G is an NSK-map if and only if
In this paper we are interested in the case where G is a cyclic group Z N .

Definition 2.4
Suppose that σ is an NEC signature (2.2) and N is a positive integer. We say that the pair (σ, N ) is admissible if there exists a NSK-map θ : → Z N with σ ( ) = σ . If, furthermore, such θ is unique up to equivalence, then we say that (σ, N ) is rigid.

Automorphisms of NEC-groups vs mapping class groups
In this subsection we recall the relationship between the outer automorphism group of an NEC-group and the mapping class group of the orbit space H/ . The outer automorphism group Out( ) is the quotient Aut( )/Inn( ) of the group of all automorphisms of by the subgroup of inner automorphisms. For simplicity we assume that the signature of has the form Set S = H/ and note that S is a non-orientable surface of genus h with r distinguished points, over which the projection p : H → S is ramified. Let P denote the set of distinguished points, U = H\ p −1 (P) and S 0 = S\P. Then p : U → S 0 is a regular covering and is its deck group isomorphic to π 1 (S 0 )/ p * (π 1 (U)). The canonical generators x 1 , . . . , x r and d 1 , . . . , d h of correspond to standard generators of π 1 (S 0 ) and p * (π 1 (U)) is normally generated by x m i i for i = 1, . . . , r . We denote by Mod(S, P) the mapping class group of S relative to P, defined as the group of isotopy classes of homomorphism of S preserving P. The pure mapping class group is the subgroup PMod(S, P) of Mod(S, P) consisting of the isotopy classes of homomorphism fixing each element of P. The groups PMod(S, P) and Out( ) are isomorphic by a generalisation, for non-orientable S, of [12, Theorem 1] (see [7,Section 3]). Given an element of PMod(S, P) one can find its image in Out( ) as follows. Represent this element by a homeomorphism f : S 0 → S 0 fixing some base point. Then f * : π 1 (S 0 ) → π 1 (S 0 ) preserves p * (π 1 (U)), hence it induces an automorphism of the quotient π 1 (S 0 )/ p * (π 1 (U)) ∼ = .

Some elementary algebra
In this paper we use additive notation for cyclic groups. For two integers a, b we denote by (a, b) their greatest common divisor, ϕ denotes the Euler's totient function. We will use the following version of Chinese Remainder Theorem. By abuse of language we will write a ∈ Z N = Z/N Z for a non-negative integer a < N . Given positive integers x and m we denote by [x] m the reduction of x modulo m.

Topological type of actions of big order
Suppose that Z N acts on a closed non-orientable surface S g of genus g ≥ 3. Then there is an NEC group and an NSK-map θ : → Z N such that g − 2 = N μ(σ ), where σ is the signature of . In particular, N > g − 2 if and only if μ(σ ) < 1. The next lemma follows by inspection. Lemma 3.1 If N > g − 2 then has one of the following signatures: where p = 2 and q arbitrary or p = 3 and q = 3, 4, 5.

Corollary 3.2 Suppose that Z N acts on a closed non-orientable surface S g of genus g ≥ 3.
If N > g − 2 then S g /Z N is one of the following orbifolds: • Klein bottle with 1 cone point, • annulus with 1 cone point, • Möbius strip with 1 cone point, • disc with 2 or 3 cone points, • projective plane with 2 or 3 cone points.
In this section we determine some rigid pairs (σ, N ) for the signatures listed in Lemma 3.1 and compute the numbers of equivalence classes of NSK-maps. The results presented here are of three types. The first type concerns necessary and sufficient conditions for (σ, N ) to be admissible -most of them follow from a more general result in [10], where such conditions are given for an arbitrary NEC signature σ . The second type of results concerns automorphisms of NEC-groups, which are related to mappings class groups of surfaces from Corollary 3.2. Some of these results are borrowed from [4] and some are new. Finally we state, in the form of corollaries, topological consequences of our results, which will play a key role in Sect. 4, and we believe that they are of independent interest.

Actions with a 1-punctured Klein bottle as the quotient orbifold
In this subsection we fix with signature (2; −; [m]; {−}). We also fix canonical generators This surface is represented on Fig. 1 as a sphere with two crosscaps, which means that the interiors of the shaded discs should be removed, and then antipodals points in each of the resulting boundary components should be identified. We have an isomorphism where x m denotes the normal closure of x m = (d 2 1 d 2 2 ) m and d 1 , d 2 are the standard generators of π 1 (S\{P}) shown on Fig. 1 (right). Let us briefly describe the generators of Mod(S, {P}) given in [14]. Consider the simple closed curve a on S shown on Fig. 1 (left). Observe that z = d 1 d 2 is represented by a simple loop freely homotopic to a. This curve is two-sided, which means that its regular neighbourhood is an annulus. Cutting S along a, twisting one of the sides by 360 • in the direction indicated by the small arrows on the figure and gluing back gives a self-homeomorphism of S, whose isotopy class is denoted by T a and called a Dehn twist along a (see [6] for a precise definition). For i = 1, 2 we denote by V i the isotopy class of a self-homeomorphism of S obtained by sliding the puncture P once along the loop v i shown on Fig. 1 (middle). These mapping classes are called puncture slides. Finally, we denote by Y the crosscap slide defined to be the isotopy class of a self-homeomorphism of S obtained by sliding the left crosscap once along a (see [14] for

Fig. 1
Klein bottle with a puncture, the curve a (left), the loops of the puncture slides (middle) and the generators of π 1 (S\{P}) (right) a precise definition). By [14,Theorem 4.9 2 T a and the generator V 1 is redundant. By computing the automorphisms of π 1 (S\{P}) induced by T a , Y and V 2 we find that they represent the same elements of Out( ) as respectively α, β and γ .
The following lemma is a particular case of Lemma 5.9 in [10] for r = 1. N ) is admissible if and only if either m divides N and N is odd, or 2m divides N and N 2m is odd.
Proof Let y and z be the generators of from Lemma 3.3 and suppose that θ : → Z N is an NSK-map. Let k denote the order of θ(z). We will see that k = m for odd N , and k = 2m for even N . By post-composing θ with an automorphism of Z N we may assume θ(z) = N /k. Set a = θ(y) and note that a and N /k must be coprime, because θ is an epimorphism. Furthermore, by pre-composing θ with a power of the automorphism α from Lemma 3.3, we may assume If N is even then k = 2m for even m, whereas for odd m either k = m or k = 2m. Since + is generated by z, yzy −1 and y 2 , θ( + ) is generated by N /k and 2a. By (3) of Lemma 2.3 these elements generate Z N , and thus N /k is odd if N is even, hence k = 2m.
Conversely, for any a ∈ Z * N /k , the mapping θ a : → Z N defined as θ a (z) = N /k and θ a (y) = a is an NSK-map. To finish the proof it suffices to show that for a, b ∈ Z * N /k , θ a and θ b are equivalent if and only if a ≡ ±b (d).
Suppose that a ≡ εb (d), where ε ∈ {1, −1} and a, b ∈ Z * N /k . Let b be the inverse of b modulo N /k. Then εb a ≡ 1 (d), and by Lemma 2.5, there exists c ∈ Z N such that c ≡ 1 (k) and c ≡ εb a (N /k). Observe that c ∈ Z * N and we have cθ b (z) = N /k and cθ b (y) = εa + l N/k for some l.
Suppose conversely, that θ a and θ b are equivalent, that is cθ b = θ a •φ for some c ∈ Z * N and φ ∈ Aut( ). By Lemma 3.3 we have θ a φ(y) = ±a +l N k for some l ∈ Z k and θ a φ(z) = ± N k . From the former equality we have cb = ±a (N /k) and from the latter one c = ±1 (k) and it follows that b = ±a (d).

Corollary 3.6 There exists an action of G = Z N on S with S/G being a Klein bottle with a single cone point of order m if and only if m and N satisfy the conditions from Lemma 3.4, and then the number of topological types of such action is ϕ(d)
2 , where d is as in Proposition 3.5. In particular, such action is unique up to topological conjugation if and only if d ∈ {1, 3}.

Actions with 1-punctured annulus or Möbius band as the quotient orbifold
Here we fix NEC groups 1  We also fix canonical generators x, e, c 1 , c 2 of 1 , satisfying the following defining relations: and canonical generators x, d, c, of 2 , satisfying the following defining relations: The following two lemmas are proved in [ ( 2 ) is isomorphic to the Klein four-group and is generated by classes of automorphisms γ , δ defined by γ : N ) is admissible if and only if N is even and m divides N .
Proof The "only if" part follows immediately for Lemma 2.3. For the "if" part, assume that 2 divides N , m divides N and define θ i : i → Z N for i = 1, 2 by Note that + i is generated by conjugates of x, e and c 1 c 2 if i = 1, and by conjugates of x and cd if i = 2. It follows from Lemma 2.3 that θ i are NSK-maps.

Remark 3.10
Similarly as a few other signatures consider in this section, the above signature σ 2 is a special case of the one from Lemma 5.14 in [10] for r = 1. Unfortunately however, there is an error in the statement of that lemma, and we take the opportunity to correct it here: namely, the condition "and some of N /2, m 1 , . . . , m r is even" must be deleted. In the proof, the authors failed to observe that c 0 d ∈ + at the very end of page 182. Consequently, assertion (iv) of Theorem 6.4 in [10] also has to be modified. Its final part should read "where α = 0 if lcm (N /N 1 , . . . , N /N r ) = N , and α = 1 otherwise." The next lemma will be crucial for the proof of Proposition 3.12. Proof (a) Note that a ∈ X if and only if a and N /m generate Z N . For a ∈ X and c ∈ H we have ca ∈ X , because c N m = N m . To see that the action is free, suppose that ca = a for a ∈ X and c ∈ H . Write c = 1 + km. We have that N divides a(c − 1), hence N m divides ak. Since   Case i = 2. Set y = cd. Every NSK-map 2 → Z N is equivalent, by multiplication by an element of Z * N , to θ a : 2 → Z N defined by θ a (x) = N m , θ a (c) = N 2 and θ a (d) = a for some a ∈ X (note that θ a ( + 2 ) is generated by N m and a, hence the condition a ∈ X is equivalent to (3) of Lemma 2.3).
For a, b ∈ X , θ a is equivalent to θ b if and only if cθ b = θ a • φ for some c ∈ Z * N and φ ∈ Aut( 2 ). By Lemma 3.7 we may assume φ ∈ {1, γ, δ, γ δ}. Then, after replacing c by −c if necessary, we have c N m = N m and either cb = a or cb = −(a + N m ). As in the case i = 1, the number of equivalence classes of NSK-maps 2 → Z N is equal to the number of conjugacy classes of orbits of the action of H on X .

Actions with a 2-punctured disc as the quotient orbifold
We fix an NEC group with signature (0; +; [m 1 , m 2 ]; {( )}) and generators x 1 , x 2 , c, satisfying the following defining relations: The following lemma is proved in [4, Proposition 4.10] Lemma 3.14 If m 1 = m 2 then Out( ) has order 2 and is generated by the class of automorphism α, defined by α : The next one is proved in [2, Theorem 3.5 and Corollary 3.3] and it is also particular case of Lemma 5.16 in [10] for r = 2.  Proof Every NSK-map θ : → Z N is equivalent (by multiplication by an element of Z * N ) to θ a defined by θ a (c) = N 2 , θ a (x 1 ) = N m 1 and θ a (x 2 ) = a N m 2 for some a ∈ Z * m 2 . We are going to show that θ a is equivalent to θ a if and only if a ≡ a (k).
Suppose that θ a is equivalent to θ a . Then θ a = bθ a φ for some b ∈ Z * N and φ ∈ Aut( ). By Lemma 3.14, for every φ ∈ Aut( ) either φ(x i ) is conjugate to ≡ 1 (m 1 ) and b a ≡ a (m 2 ), hence a ≡ a (k).
Conversely, suppose that a ≡ a (k). By Lemma 2.5, there exists unique b ∈ Z * N such that b ≡ 1 (m 1 ) and b ≡ (a ) −1 a (m 2 ), where (a ) −1 is the inverse of a in Z * m 2 . We have θ a = bθ a .
To finish the proof it suffices to note that for each d ∈ Z * k there exists a ∈ Z * m 2 such that d ≡ a (k). Hence, equivalence classes of NSK-maps → Z N are in one to one correspondence with elements of Z * k .

Actions with a 2-punctured projective plane as the quotient orbifold
In this subsection we fix with signature (1; −; [m 1 , m 2 ]; {−}) and canonical generators x 1 , x 2 , d of , satisfying the following defining relations:

Lemma 3.19
If m 1 = m 2 , then Out( ) is isomorphic to the Klein four-group and is generated by classes of automorphisms α, β defined by α : Proof Recall from Sect. 2.2 that Out( ) is isomorphic to the pure mapping class group PMod(S, {P 1 , P 2 }), where S = H/ a projective plane with 2 distinguished points P 1 , P 2 ∈ S (Fig. 2). For i = 1, 2 let V i denote the isotopy class of the self-homeomorphism of S obtained by sliding the puncture P i once along the loop v i on Fig. 2. By [14,Corollary 4.6] PMod(S, {P 1 , P 2 }) is generated by V 1 and V 2 and is isomorphic to Z 2 × Z 2 . By computing the automorphisms of π 1 (S\{P 1 , P 2 }) induced by V 1 and V 2 we find that they represent the same elements of Out( ) as respectively β and α.
The next lemma is a special case of [10, Lemma 5.8] for r = 2.   that (σ, N ) is admissible, N is odd, m 1 = m 2 and k = (m 1 , m 2 ). There are exactly ϕ(k) 2 equivalence classes of NSK-maps → Z N .
Proof Every NSK-map θ : → Z N is equivalent (by multiplication by an element of Z * N ) to θ a such that θ a (x 1 ) = N m 1 and θ a ( relation 2θ a (d) = −(θ a (x 1 ) + θ a (x 2 )). Similarly as in the proof of Proposition 3.16 it can be shown that θ a is equivalent to θ a if and only if a ≡ ±a (k) (the only difference is that now admits an automorphism, e.g. α from Lemma 3.19, such that α(x 1 ) and α(x 2 ) are conjugate respectively to x 1 and x −1 2 ). If k > 1, then it is impossible that a ≡ −a (k) for a ∈ Z * m 2 , because k is odd. Hence, there are ϕ(k) 2 equivalence classes of NSK-maps if k > 1, and one class if k = 1. that (σ, N ) is admissible, N is even, m 1 = m 2 and k = (m 1 , m 2 ). There are exactly ϕ(k) equivalence classes of NSK-maps → Z N .

Proposition 3.22 Suppose
Proof Let θ : → Z N be a NSK-map. After multiplication by an element of Z * N we may assume that θ(x 1 ) = N m 1 and θ(x 2 ) = a N m 2 for some a ∈ Z * m 2 . We have 2θ(d) + θ(x 1 ) + θ(x 2 ) = 0, and since N is even, θ(d) is determined by θ(x 1 ) and θ(x 2 ) only modulo N 2 . Suppose that θ : → Z N is another NSK-map, such that θ (x 1 ) = N m 1 and θ (x 2 ) = a N m 2 for some a ∈ Z * m 2 . We claim that θ and θ are equivalent if and only if either 1. a ≡ a (k) and θ (d) = bθ(d), where b is the unique element of Z * N satisfying b ≡ 1 (m 1 ) and ba ≡ a (m 2 ), or 2. a ≡ −a (k) and θ (d) = b(θ (d)+θ(x 1 )), where b is the unique element of Z * N satisfying b ≡ −1 (m 1 ) and ba ≡ a (m 2 ).
Suppose k > 2. It follows from the previous paragraph that there is a surjection ρ from the set of equivalence classes of NSK-maps onto Z * where θ is as above, [θ ] is its equivalence class. We note that ρ is 2-to-1. Indeed, take θ and θ as above and suppose that they are not equivalent, but ρ[θ ] = ρ[θ ]. Then either (1') a ≡ a (k) and θ (d) = bθ(d) + N 2 , where b is as in (1) above, or (2') a ≡ −a (k) and θ (d) = b(θ (d) + θ(x 1 )) + N 2 , where b is as in (2) above.
In case (1 ), θ is equivalent (by multiplication by b −1 ) to θ 1 defined by θ 1 ( In case (2 ) θ is equivalent (by multiplication by −b −1 ) to θ 2 defined by θ 2 ( Observe that θ 1 and θ 2 are equivalent [by (2) We have to show that θ and θ are equivalent. Let b be the unique element of Z * N such that b ≡ −1 (m 1 ) and b ≡ 1 (m 2 ). By (2) it suffices to show that It follows that all NSK-maps → Z N are equivalent.
where k = (m 1 , m 2 ). In particular, such action is unique up to topological conjugation if and only if k ≤ 3.
α : There exists an action of G = Z N on S with S/G being a projective plane with 3 cone points of orders 2, p, q if and only if p, q and N satisfy the conditions from Lemma 3.27, and such action is unique up to topological conjugation.

Self-homeomorphisms with large periods
Let C(g, N ) denote the number of topological types of action of Z N on a closed non-orientable surface of genus g ≥ 3. It is proved in [2] that C(g, N ) = 0 for N > 2g, and if g is even then C(g, N ) = 0 for N > 2(g − 1). In this section we compute C(g, N ), for all g ≥ 3 and N > max{g, 3 2 (g − 2)}. For an admissible pair (σ, N ), let c(σ, N ) denote the number of equivalence classes of NSK-maps θ : → Z N , where is an NEC-group with σ ( ) = σ (see Sect. 2 for definitions). Then C(g, N ) is the sum of all c(σ, N ) such that (σ, N ) is an admissible pair satisfying N μ(σ ) = g − 2.
By Lemma 3.20 a pair (σ 3 , N ) is admissible if and only if N = m is even and m/2 is odd. Every such pair is rigid by Proposition 3.22. We have N = 2(g − 1) and g is even.
By Lemma 3.4 a pair (σ 12 , N ) is admissible if and only if 4 divides N and 8 does not divide N . Such pair is rigid by Proposition 3.5. We have N = 2(g − 2) and 4 divides g.
Thus, the only possible values for N are those given in the theorem. For each of these values we calculate C(g, N ) by adding up c(σ, N ) for all admissible pairs (σ, N ) from Table 1.