Fixed Points on Asymmetric Riemann Surfaces

We study actions of cyclic groups on asymmetric Riemann surfaces for which all conformal automorphisms of prime orders have the same number of fixed points. We find the sharp upper bound on the number of points fixed by the square of an anticonformal automorphism of an asymmetric surface. Moreover, we determine the minimal genus of an asymmetric Riemann surface on which a given finite group G acts without fixed points.


Introduction
In this paper, we study the so-called asymmetric (or pseudo-real, see [1]) Riemann surfaces. This term means that the surface admits orientation reversing automorphisms, but none of them is an involution. Such automorphisms are called asymmetries and they have orders divisible by 4. An asymmetry of order 4 is called a pseudo-symmetry and the underlying Riemann surface is called pseudo-symmetric. Let us also mention that an orientation reversing involution is called a symmetry, and the surface on which it acts is called symmetric.
The category of complex algebraic curves is equivalent to the category of compact Riemann surfaces. It is known that the moduli space M g of complex algebraic curves of genus g is a quasi-projective variety which can be defined in P n (C) by polynomials with rational coefficients. There is an antiholomorphic involution ι : M g → M g which maps the class of a complex curve to its conjugate. The fixed points of such a mapping are called complex algebraic curves with real moduli. The defining equations of symmetric Riemann surfaces are over reals and therefore they are fixed by ι. The other fixed points, which are not symmetric, correspond to the asymmetric surfaces.
There are many publications concerning asymmetric and pseudosymmetric Riemann surfaces, let us only mention the most important, from our point of view, which were the motivation for this work. Many general results concerning asymmetric surfaces can be found in [1] and some asymmetric surfaces with cyclic automorphism groups were also considered in [4]. The hyperelliptic asymmetric surfaces were studied in [12] and in [2], where the defining equations for such surfaces are given and the special case of hyperelliptic pseudo-symmetric surfaces is also treated. The present paper can be seen as a continuation of the authors' papers [8,9,13], where asymmetric p-hyperelliptic and (q, n)-gonal surfaces, with cyclic automorphism group, were studied.
We shall also recall the two papers concerning fixed points on Riemann surfaces, which are related to the main theme of this work. In [5] the number of fixed points of the product of two symmetries of a Riemann surface is studied, while here we study the number of points fixed by the square of an asymmetry. Also, in [7] the first author of this paper considered the fixed point free actions on symmetric surfaces, while here in the last part of the paper we study similar problems in the asymmetric case.
In this paper, we study fixed points of conformal automorphisms of asymmetric Riemann surfaces of genus g > 1. We use the combinatorial approach to the problem based on the uniformization theorem and on the theory of non-euclidean crystallographic groups, called shortly NEC groups.
The second chapter provides some information about NEC groups necessary for understanding the topic, and it presents two well-known theorems about fixed points. The first proved by Macbeath in [10], gives a formula on the total number of points fixed by a conformal automorphism of a Riemann surface. The second one proved by Izumi in [6] gives the upper bound on this number.
In the third chapter we use the necessary and sufficient conditions for the existence of an asymmetric Riemann surface of genus g with the cyclic automorphism group Z 4k = δ , given in [13], in order to determine all possible integers which can be realized as the numbers of fixed points of δ 2 . In particular, we find the upper bound on this number and we study its coincidence with Izumi's bound.
For a given pair (g, k), let r = g mod 2k − 1, c = (g − r)/(2k − 1), r (k) = r mod k and g (2) = g mod 2. We prove that if g > 2k − 1 and g (2) ≤ r (k) ≤ c + 1, then there is an asymmetric Riemann surface of genus g having an asymmetry δ of order 4k, whose square has exactly 2(c + 1 − r (k) ) fixed points. Later on, we find the minimal number of topologically nonequivalent actions of the group δ . The fourth chapter is devoted to the so-called fixed point stable actions of the automorphism group G on an asymmetric Riemann surface. We say that such an action is β-fixed point stable if all conformal automorphisms of prime order in G fix exactly 2β + 2 points for an integer β > −1. If this condition is satisfied by all such conformal automorphisms except involutions, then we say that the action is β-exceptional fixed point stable. We prove that for given integers β and k, there is an infinite sequence (g t ) corresponding to β-fixed point stable actions of the group Z 4k on asymmetric Riemann surfaces of genera g t . We find the necessary and sufficient conditions for the existence of an asymmetric pseudo-symmetric Riemann surface with β-fixed point stable action of the group Z 4k , and we determine the smallest genus of such a surface. In the fifth chapter we find the minimal number of βexceptional fixed point stable actions of the group Z 4k on an asymmetric Riemann surface of genus g.
In the last part of the paper, we study fixed point free actions for which none automorphism of an asymmetric surface has fixed points. We construct such actions with a given finite group G standing as the group of conformal automorphisms of the surface and we find the lower bound on its genus. In particular, we prove that any nilpotent group of even order acts without fixed points on an asymmetric Riemann surface.

Preliminaries
We shall use the combinatorial approach, based on Riemann uniformization theorem and non-euclidean crystallographic groups. Recall that such a NEC group is just a discrete and cocompact subgroup of the group G of isometries of the hyperbolic plane H, including those which reverse orientation, and if such a subgroup contains only orientation preserving isometries, then it is called a Fuchsian group. For every NEC group Λ, we have the associated signature σ(Λ), which determines its algebraic structure. It has the form The numbers m i ≥ 2 are called the proper periods, the non-empty brackets C i = (n i1 , . . . , n isi ), i = 1, . . . , u, or the empty ones (−), are the period cycles, the numbers n ij ≥ 2 are the link periods and h ≥ 0 is said to be the orbit genus of Λ. The orbit space H/Λ is a surface of topological genus h, having u + v boundary components, and being orientable or not according to the sign being + or −. The group with the signature (1) has the presentation given by the following generators and relations: A Fuchsian group can be regarded as a NEC group with the signature usually shortened to (h; m 1 , . . . , m t ); a Fuchsian group without periods is called a Fuchsian surface group.
Any set of generators of a NEC group satisfying the above relations is called a canonical set of generators. Every NEC group has an associated fundamental region, whose hyperbolic area μ(Λ), for a NEC group Λ with signature (1), is given by where ε = 2 if the sign is + and ε = 1 otherwise. Since the above formula depends on the signature of the group, sometimes we shall write μ(σ) instead of μ(Λ) when we refer to the hyperbolic area of a fundamental region of a NEC group Λ with signature σ. Now, if Γ is a finite index subgroup in a NEC group Λ, then it is a NEC group itself and there is a Hurwitz-Riemann formula, which says that Finally, it is known that an abstract group with the presentation (2) can be realized as a NEC group Λ with the signature (1) if and only if the quantity in (4) is positive.
Every compact Riemann surface X of genus g ≥ 2 is isomorphic to the orbit space H/Γ for some surface Fuchsian group Γ. A finite group G is a group of automorphisms of X if and only if G = Λ/Γ for some NEC group Λ containing Γ as a normal subgroup. In such a case we say that G acts on X with signature σ, where σ denotes the signature of Λ. If Λ is a maximal NEC group, then there does not exist another NEC group containing it properly, and the group G is the full automorphism group of X. The detailed exposition of maximality can be found in [3]. A signature σ is called maximal, if for every NEC group Λ with a signature σ containing a NEC group Λ with the signature σ and having the same Teichmüller dimension, the equality Λ = Λ holds. For any maximal signature σ, there exists a maximal NEC group with the signature σ.
In the paper we will use the two well-known facts about the number of points fixed by a conformal automorphism of a Riemann surface. The first is the formula on the total number of such points given by Macbeath in [10].

The Number of Points Fixed by the Square of an Asymmetry
Let δ be an orientation reversing automorphism of an asymmetric Riemann surface X of genus g ≥ 2. We shall study the number of fixed points of the conformal automorphism δ 2 . Let us notice that δ has order divisible by 4 because if d = (δ) was odd, then δ would be conformal as a power of δ 2 while if d ≡ 2 mod 4, then δ d/2 would be a symmetry. So assume that d = 4k for some integer k ≥ 1. Since X has no symmetries, it follows that G = δ acts with a signature for some nonnegative integers l, h and some proper divisors m 1 , . . . , m s of 2k.
In [13] the second author found the necessary conditions for the existence of a smooth epimorphism from a NEC group with the signature (6) onto Z 4k and justified that this signature must be maximal. As the result we have the following for some nonnegative integers h, l and some proper divisors m 1 , . . . , m s of 2k such that (i) h has the same parity as g, . . , s}, (iii) (h, s, l) = (1, 1, 0), (0, 1, 1) and (0, 2, 0).
In order to simplify the notation, we will write a (b) instead of a mod b for natural numbers a and b. In the case when a Riemann surface X has an automorphism of order 4k, we will present the genus of X in the form g = c(2k − 1) + r for c ≥ 0 and 0 ≤ r ≤ 2k − 2. Throughout the paper we shall use the following symbols: Moreover, for any group G, by [g 1 , g 2 ] we denote the commutator of two elements g 1 , g 2 ∈ G defined by the formula [g 1 ,

Conversely, for any R satisfying the last inequalities, there exists such an automorphism δ if and only if
for an integer h ≥ 0 of the same parity as g and some proper divisors Thus by Theorem 2.1, δ 2 has f (δ 2 ) = 2l fixed points. By (7), Thus f (δ 2 ) ≤ 2( g 2k−1 + 1) which is consistent with Theorem 2.2. Using the notation (8), we can write l in the form l = c + 1 − R for By the last formula, r is equal to (9). Since 0 ≤ l ≤ g 2k−1 + 1, it follows that R must be a nonnegative integer not exceeding c + 1.
Conversely, let R be an integer in the range 0 ≤ R ≤ c + 1. If there exist h ≥ 0 of the same parity as g and some proper divisors m 1 , . . . , m s of 2k satisfying the conditions (ii) and (iii) of Lemma 3.1 for which r can be expressed by (9), then according to Lemma 3.1, there is an action of the group G = Z 4k = δ with the signature on a Riemann surface X of genus g = c(2k−1)+r and δ 2 has 2(c+1−R) fixed points. By the condition (iii), the signature is maximal and so we can assume that G is the full automorphism group. Thus X is asymmetric because it has an anti-conformal automorphism δ, but it has no symmetries.
We will say that an anti-conformal automorphism δ of an asymmetric Riemann surface X of genus g is a [k, l, g]-asymmetry, if (δ) = 4k and δ 2 fixes exactly 2l points on X.
and there is an integer γ which, depending on r, satisfies the following conditions: (2) in the case (B).
According to Theorem 3.2, there exists a [k, l, g]-asymmetry for l = c + 1 − R, if the selected above parameters satisfy the equality (9) and the conditions (ii) and (iii) of Lemma 3.1. It is easy to check that the condition (iii) fails only in few cases at which g ≤ 2k − 1. Suppose that the condition (ii) is not satisfied. Then g must be even and γ = c + 1. However, in the case (A) for g > 2k − 1, there are at least four periods equal to 2. Thus we can reduce the number of such periods by 4 and increase the number h by 2 at the same time. After this change the condition (ii) is satisfied. If k is odd and γ = k − r in the case (B), then there are periods 2 and k for which the condition (ii) is also true. (2) , and the upper bound is sharp if and only if 2k + g (2) − r divides 2k.
Proof. According to Theorem 3.2, l = c + 1 − R for some nonnegative integer R for which there exist h of the same parity as g and some proper divisors m 1 , . . . , m s of 2k such that r is given by (9). Thus an upper bound on l equals c + 1. We will find the conditions on g for which this bound is sharp. If R = 0, then r given by (9) is in the range 0 ≤ r ≤ 2k − 2, if and only if (h, s) = (0, 0) or (h, s) = (0, 1). Thus r = 0 or r = 2k − 2k m1 . In both cases 2k − r divides 2k, and g is even because it must have the same parity as h. Consequently, R ≥ 1  1). Thus r = 1 or r = 1 + 2k − 2k m1 . In both cases g is an odd integer for which 1 + 2k − r divides 2k. Summing up, l ≤ c +1− g (2) and if l attains this upper bound, then 2k − r + g (2) divides 2k. Conversely, if the last condition is satisfied, then applying Corollary 3.3 for γ = g (2) we get a [k, l, g]-asymmetry for which l attains the upper bound c + 1 − g (2) .
and this bound is sharp.
There are some Riemann surfaces corresponding to complex algebraic curves with real moduli for which the Izumi's bound is sharp. In [5] G. Gromadzki and the first author proved that the maximal number 2(g+n−1) n−1 of fixed points is attained for the product of two symmetries, which generate a dihedral automorphism group D n of a symmetric Riemann surface of genus g. In the asymmetric case, we have the following result. Proof. By the proof of Theorem 3.2, f (δ 2 ) = 2l for l given by (10). Thus f (δ 2 ) = 2( g 2k−1 + 1) if and only if g is divisible by 2k − 1 and h = s = 0. Hence the group δ acts with the signature By Lemma 3.1, this action is possible only for even g because h must have the same parity as g. So g ≡ 0 (4k − 2). If δ is a pseudo-symmetry, then the signature (13)  Proof. Suppose that r ≥ k. Then applying Corollary 3.3 for γ = r (k) , we conclude that there exists a [k, c + 1 − r (k) , g]-asymmetry. If r < k, then for R = r (k) , there are parameters h = g (2) , s = 2(r (k) − g (2) ) and m 1 = · · · = m s = 2 satisfying the conditions of Theorem 3.2, and so such an asymmetry exists also in this case. In both cases the group generated by the asymmetry acts with the signature We say that a pair (g, k) of integers g, k ≥ 2 satisfies the cyclic action conditions, if there exists a [k, l, g]-asymmetry δ for some integer l in the range 0 ≤ l ≤ c + 1 − g (2) , where c = g 2k−1 . In such a case there are parameters h, m 1 , . . . , m s satisfying the conditions of Lemma 3.1, and δ 2 fixes 2l = 2(c + 1 − R) points for R given by (11). Among many choices of these parameters, there is one for which R has the smallest possible value. Then the number of points fixed by δ 2 attains the upper bound which is denoted by f k,g max .
Proposition 3.8. Suppose that a pair (g, k) satisfies the cyclic action conditions. Let n 1 , . . . , n p be all proper divisors of 2k, and let e 1 , . . . , e p be nonnegative integers such that whose square has f k,l max − 4ek/n fixed points. Repeating this construction for all proper divisors of 2k in such a way that the conditions of Lemma 3.1 are satisfied, we obtain a [k, l, g]-asymmetry whose square has 4kei ni fixed points.

Fixed Point Stable Actions on Asymmetric Surfaces
A Riemann surface X is called (q, n)-gonal if it admits a conformal automorphism ρ of prime order n such that X/ ρ has genus q. The cyclic group G = δ = Z 4k acts on an asymmetric Riemann surface with a signature (6). Thus by (5) Hence by Theorem 2.1, any conformal automorphism in G has an even number f (ρ) of fixed points. We will say that an action of G is β-fixed point stable if all automorphisms of prime orders have the same positive number of fixed points equal to 2β +2 for some integer β > −1. When this condition is satisfied by all such automorphisms except involutions, we will say that an action is β-exceptional fixed point stable.
Lemma 4.1. If an action of the group G = Z 4k on an asymmetric Riemann surface X of genus g is β-fixed point stable, then g + β ≡ 0 (2k), and if it is β-exceptional fixed point stable, then g + β ≡ 0 (k/2 t ) for the greatest power 2 t dividing k. Moreover, the surface X is a n-sheeted covering of an orbifold of genus g+β n − β, ramified over 2β + 2 fixed points, for any prime n dividing 2k, which is different from 2 in the exceptional case.
Proof. Assume that G = Z 4k acts with signature (6) on an asymmetric Riemann surface X of genus g, and let ρ ∈ G be a conformal automorphism of prime order n. Then ρ has an even number f (ρ) of fixed points and by the Hurwitz-Riemann formula, the group ρ acts with the signature for β = f (ρ)/2 − 1. In particular, the surface X is ( g+β n − β, n)-gonal and β satisfies the conditions: Let D be the set of indices i in the signature (6) for which n divides m i and let N be the set of indices for which this is not true. Then by Theorem 2.1, β + 1 = i∈D 2k mi + l. By substituting the last into (7), we get Thus g + β ≡ 0 (n a ) for the greatest power n a dividing 2k. For a fixed point stable action, this congruence is true for all prime n dividing 2k, which implies g + β ≡ 0 (2k). For an exceptional fixed point stable action, we must exclude n = 2 and so g + β ≡ 0 (k/2 t ) for the greatest power 2 t dividing k.

Proposition 4.2.
For given β ≥ 0 and k ≥ 2, there is an infinite sequence (g t ) corresponding to β-fixed point stable actions of the group Z 4k on asymmetric Riemann surfaces of genera g t .
By Theorem 2.1, this action is β-fixed point stable.
For even k, the conditions (19) are sufficient for the existence of such an action on an asymmetric Riemann surface, which is not pseudo-symmetric.
Proof. An asymmetric Riemann surface with the automorphism group Z 4k is pseudo-symmetric if and only if k is odd. If an action of the group Z 4k with odd k is β-exceptional fixed point stable, then by for an integer y in the range 0 ≤ y ≤ β+β (2) 2 and an integer h of the same parity as g such that 0 ≤ h ≤ (g + β)/2k + (y − β). If g > 2k − 1, then for any pair (h, y) the condition (iii) of Lemma 3.1 is satisfied. The condition (ii) of this Lemma may not be satisfied for (h, y) = (0, β+1 2 ) in the case when g is even and β is odd. However, in this case we can choose (h, y) = (0, 0). So there is at least one action of the group Z 4k with the signature (20) on an asymmetric Riemann surface of genus g > 2k − 1. By Theorem 2.1, this action is β-exceptional fixed point stable.

Classes of Asymmetric Surfaces with a Cyclic Automorphism Group
An action of a finite group G on a Riemann surface of genus g ≥ 2 corresponds to a short exact sequence of homomorphisms where Λ is a NEC group and Γ is a surface Fuchsian group isomorphic to the fundamental group of the surface. Let us denote such an action by (Λ, θ, G). (21) Proof. Let g = c(2k − 1) for c > 2. Then for any integer t in the range 0 t c+1 2 −g (2) and an even nonnegative s not exceeding t the parameters in the signature (1 + g (2) + s; −; [k, t+g (2) . . . , k, 2, 2(t−s) . . . , 2, 2k, c+1−2g (2) satisfy all conditions of Theorem 3.1. Hence for any such pair (s, t), there is an action of the group Z 4k on an asymmetric Riemann surface of genus g with signature (22), and according to Theorem 2.1, this action is (c − 2g (2) )-exceptional fixed point stable. For a given t, the number n(t) of all nonnegative even integers not exceeding t is equal to t+1 2 or t 2 + 1 according to whether t is odd or even. Thus the total number N of all actions with signatures (22) is equal to  (2) is odd or even. Hence we get (21).
Let ε = 1 in the case when g and k are even and β is odd, and let ε = 0 in the remaining cases. Then there are at least N topologically non-equivalent β-exceptional fixed point stable actions of the group G = Z 4k on asymmetric Riemann surfaces of genus g, where according to whether b is odd or even, respectively.
Proof. By Lemma 3.1, there is an action of the group G = Z 4k on an asymmetric Riemann surface X of genus g with the signature Let Λ and Γ be NEC groups such that X = H/Γ and G = Λ/Γ. Since σ 0 is a maximal signature, we can choose Λ as a maximal group, which is not contained properly in any other NEC group. Then G is the full automorphism group of X. By the Hurwitz-Riemann formula, A = kμ(Λ)/π = g − 1, where μ(Λ) is given by (4). Actions of the group G with signatures which have different sets of proper periods or different orbit genera are not topologically equivalent. Thus all changes of parameters in σ 0 , for which the integer A remains unchanged, provide different classes of surfaces in M g with automorphism group Z 4k . We cannot reduce l = β + 1 by 1, because there is no change of other parameters in the signature, which could compensate for this. However, we can reduce l by 2t for any integer t in the range 0 ≤ t ≤ l 2 , by adding new periods 2, 2t . . ., 2, k t . . ., k at the same time. The parameter h = g (2) in σ 0 has the smallest possible value. It can increase only by an even number because it must have the same parity as g. We can increase h by an even number s through reducing by 2s the number of periods equal to 2, under condition that there are at least 2s of such periods in the signature. As a result, we get topologically non-equivalent actions of the group Z 4k with signatures where t is an integer in the range , and s is an even nonnegative integer not exceeding a + t for a = g+β 2k − g (2) − β. We check that if g > 2k − 1, then the condition (iii) of Lemma 3.1 is satisfied for any pair (t, s). The condition (ii) of Lemma 3.1 is not satisfied if and only if g and k are even and (t, s) = (b, 0) for an odd β. depending on whether a + t is even or odd. In the case when g and k are even and t = b for odd β, we must reduce the number n(t) by 1 in order to reject s = 0. The number N of all signatures of the form (26) is equal to N = b t=0 n(t). By considering the four cases depending on the parity of a and b we calculate that N is equal to (23) or to (24) according to whether b is odd or even, respectively. Corollary 5.3. For any integer g = c(2k − 1) + r such that g > 2k − 1 and g (2) ≤ r (k) ≤ c + 1 there are at least N topologically non-equivalent actions of a cyclic automorphism group G = Z 4k on asymmetric Riemann surfaces of genus g, for N given by (23) or (24) with a = r (k) − g (2) , according to is odd or even, respectively.
Proof. This is a particular case of Theorem 5.2 with β = c − r (k) . Now the signatures (26) have the form . Thus the number of non-equivalent actions cannot be smaller than the number of different choices of parameters t and s for which the conditions of Lemma 3.1 are satisfied. The last number is equal to (23) or (24), according to the parity of b, where a = r (k) − g (2) and b = c+1−r (k) 2 . Let us notice that for t = s = 0 we get an action with the signature (14).

Corollary 5.4. Let M
(q,n) g ⊂ M g be the locus consisting of classes of surfaces admitting a (q, n)-gonal automorphism. For a given pair (g, k) such that g ≥ (k − 1)(2k − 1), let N be an integer given by (21) if g (2k−1) = 0, and in the remaining cases let N be given by (23) or (24) for a = r (k) − g (2) and b = c+1−r (k) 2 , according to whether b is odd or even. Then there are at least N topologically non-equivalent actions of the group Z 4k on surfaces in n M (q,n) g , where n runs over all odd prime integers dividing k, and q = g+β n − β for β = c − 2g (2) if g (2k−1) = 0, and β = c − r (k) in the remaining cases.
Proof. Suppose that G = Z 4k acts with the signature (6) and let ρ ∈ G be a conformal automorphism of a prime order n. Then the number of points fixed by ρ is equal to f (ρ) = 2(l +

Fixed Point Free Actions on Asymmetric Riemann Surfaces
Let G ± be the group Aut ± (X) of conformal and anti-conformal automorphisms of an asymmetric Riemann surface X. Let us notice that an automorphism of an asymmetric surface may have fixed points only if it is conformal. We will say that an action of the group G ± is fixed point free if none of its nontrivial conformal automorphisms has fixed points.
Theorem 6.1. Let X be an asymmetric Riemann surface of genus g with a fixed point free action of the group G ± = Aut ± (X), and let s be the minimal number of generators of the subgroup G + < G ± consisting of all conformal automorphisms. Then g is an odd integer such that g ≥ 1 + |G + | if s ≤ 4 and Proof. Let Λ and Γ be NEC groups for which X H/Γ and G ± = Λ/Γ. The group Γ is a surface Fuchsian group with the signature (g; +; [−]; {−}), where g ≥ 2 is the genus of X. Since X has no symmetries, it follows that there can be no period cycles in the signature of Λ. Moreover, by Theorem 2.1, there can be no proper periods in it. Thus Applying the Hurwitz-Riemann formula for (Λ, Γ), we get Thus g ≡ 1 mod 2, because G ± has an anticonformal automorphism whose order is a multiple of 4. Let s denote the minimal number of generators of the subgroup G + < G ± consisting of all conformal automorphisms. Then G + = Λ + /Γ for the canonical Fuchsian subgroup Λ + of Λ. By (5), Λ + has the signature (γ − 1; +; [−]; {−}). Thus G + is generated by 2γ − 2 elements, being the images in G + of the 2γ − 2 generators of the group Λ + . In particular, 2γ − 2 ≥ s and by (29) we get g ≥ 1 + |G ± | 4 (s − 2). The group G ± cannot act with signature (28) for γ = 1, 2, because then μ(Λ) ≤ 0. For γ = 3, this signature is non-maximal and there is a NEC group Λ with the signature (0; +; [2,2,2]; {(−)}) which contains Λ as a subgroup of index 2. In this case, Λ is generated by elements x 1 , x 2 , x 3 , e, c which satisfy the relations x 2 i = 1 for i = 1, 2, 3, x 1 x 2 x 3 e = 1 and e −1 cec = 1, and an embedding i : Λ → Λ is given by where d 1 , d 2 , d 3 denote canonical glide reflections of Λ.
Let θ : Λ → G ± be an epimorphism with kernel Γ. Then G ± is generated by three anti-conformal elements h 1 , h 2 and h 3 which satisfy the relation h 2 1 h 2 2 h 2 3 = 1. No composition of an odd number of these elements has order 2 because X has no symmetries. Suppose that there is a semidirect product G = G ± ρ : ρ 2 = 1 for which θ can be extended to an epimorphism θ : Λ → G with kernel Γ. Since there is no anti-conformal automorphism of order 2 in G ± , it follows that θ (c 0 ) = ρ. By the equality d 3 = x 3 c, we get θ (x 3 ) = h 3 ρ. The surface Fuchsian group Γ has no elliptic elements and therefore the last element has order 2 which implies that ρh 3 ρ = h −1 3 . Next, by the equality d 2 = x 2 x 3 cx 3 we get θ (x 2 ) = h 2 h 2 3 ρ and this element has order 2 if and only if ρh 2 Similarly, by the equality So the existence of an automorphism η ∈ Aut(G ± ) induced by is the necessary and sufficient condition for the existence of an extension G = G ± ρ of G ± . Then θ can be extended to an epimorphism θ : Λ → G for which θ • i = j • θ, where j : G ± → G is the identity. This stands in contradiction to the assumption that G ± is the full automorphism group of all conformal and anti-conformal automorphisms of the surface X. Moreover, in this case X is not asymmetric because G has a symmetry ρ. However, if G ± has not the epimorphism η, then an action of G ± with the signature (3; −; [−]; {−}) on an asymmetric Riemann surface of genus g = 1+|G ± |/2 = 1 + |G + | is still possible and this is the lower bound on g for s ≤ 4.
Theorem 6.2. The lower sharp bound on the genus g of an asymmetric Riemann surface, whose full automorphism group G ± has no fixed points, is equal to 1 + |G ± | or 1 + |G ± | 2 , according to whether G ± is abelian or non-abelian. This bound is attained for infinitely many values of g.
Proof. The automorphism group G ± of an asymmetric Riemann surface of genus g ≥ 2 acts without fixed points with a signature σ γ = (γ; −; [−]; {−}) for γ ≥ 3. Thus according to the Hurwitz-Riemann formula, g = 1+ |G ± | 2 (γ − 2). By the proof of Theorem 6.1, an action with the signature σ 3 is possible under condition that η : G ± → G ± , induced by (30), is not an automorphism, which implies that G ± is not abelian. There are infinite families of non-abelian groups acting on asymmetric Riemann surfaces with the signature σ 3 . First, we will give one of them as an example. Let G be the 33-rd group of order 64 in the Small Groups Database. It is generated by two elements g 1 and g 2 of order 4, which satisfy the relations For a NEC group Λ with the signature σ 3 , there is a smooth epimorphism θ : Λ → G given by Hence the lower bound on g is attained for infinitely many values of g.
In the abelian case the lower bound g ≥ 1 + |G ± | is sharp because by Lemma 3.1, for any g satisfying the congruence g ≡ 1 (4), there is an action of the cyclic group Z g−1 = δ with the signature σ 4 on an asymmetric Riemann surface of genus g.
In order to prove that a given finite group G is the group of orientation preserving automorphisms of an asymmetric Riemann surface X and the action of the group Aut ± (X) is fixed point free, we will construct some group G containing G as a subgroup of index 2 and we will prove that there is an action ofG on X with a signature (28). In this case, we will use the following. Remark 6.3. Let θ : Λ →G be an epimorphism from a maximal NEC group Λ with a signature (28) onto a finite groupG such that Γ = kerθ is a surface Fuchsian group. Then the group of all conformal and anti-conformal automorphisms of the Riemann surface X = H/Γ is isomorphic toG. The surface X is asymmetric if there is no symmetry inG. Moreover, a subgroup G <G of index 2 whose generators can be written as products of even numbers of θ-images of the canonical glide reflections of Λ consists of all conformal automorphisms of X.
Proof. The groupG is isomorphic to the quotient group Λ/Γ acting on the Riemann surface X = H/Γ. Since Λ is a maximal NEC group, it follows thatG cannot be contained properly in any other automorphism group of X, because otherwise there would be a NEC group Λ containing Λ as a proper subgroup, what is impossible. HenceG is the group Aut ± (X) of all conformal and anticonformal automorphisms of the surface. It is generated by automorphisms h i = θ(d i ) for 1 ≤ i ≤ γ, where d i are the canonical generating glide reflections of the group Λ. So the surface X has anticonformal automorphisms and it is asymmetric if none of them is a symmetry.
Let G <G be a subgroup of index 2 whose generators can be written as products of even numbers of elements h i . Then G is generated by conformal automorphisms and so it is contained in the group G + of all conformal automorphisms of X. Since |G| = |G + |, it follows that G = G + . Theorem 6.4. Let G be a finite group whose center Z(G) has even order. Then there exists an asymmetric Riemann surface X, whose Aut ± (X) acts without fixed points and G is the subgroup of all conformal automorphisms in Aut ± (X).
Proof. Let G be a finite group which has a central element a of even order 2k and let C = δ be a cyclic group of order 4k. Then the direct product G × C has a normal subgroup generated by b = δ 2 a −1 andG = (G × C)/ b 131 Page 18 of 22 E. Kozłowska-Walania and E. Tyszkowska MJOM has order 2|G|. Let {g 1 , . . . , g s } be a generating set of G with g 2 i = a −1 for all i. The signature (28) with γ = 2s + 2 is maximal. Let Λ be a maximal NEC group Λ with this signature, and let θ : Λ →G be an epimorphism given by θ(d 2i−1 ) = δg i , θ(d 2i ) = (δg i ) −1 for 1 ≤ i ≤ s θ(d 2s+1 ) = δ and θ(d 2s+2 ) = δ −1 . Then θ(d 1 ) 2 · . . . · θ(d γ ) 2 = 1 and so the kernel Γ = kerθ is a surface Fuchsian group. By Remark 6.3,G is isomorphic to the group G ± = Λ/Γ of all conformal and anticonformal automorphisms of the Riemann surface X = H/Γ, which by the Hurwitz-Riemann formula has genus 2|G|s + 1. It is generated by reversing orientation automorphisms θ(d i ) for 1 ≤ i ≤ γ and its subgroup G + of all conformal automorphisms consists of elements which are products of an even number of some generators θ(d i ). In particular, the group G is contained in G + . Since |G| = |G + |, it follows that these groups are equal. Finally, the surface X is asymmetric because it has anticonformal automorphisms but none of them is a symmetry. Corollary 6.5. For any nilpotent group G of even order there exists an asymmetric Riemann surface X on which G acts as the group of all conformal automorphisms and Aut ± (X) acts without fixed points.