On dimensions of the real nerve of the moduli space of Riemann surfaces of odd genus G

In the moduli space Mg of Riemann surfaces of genus g ≥ 2 there is important, so-called, real locus Rg, consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. Rg itself is covered by the strata Rg , each being formed by the points corresponding to surfaces having a symmetry of given topological type k. These strata are known to be real analytic varieties of dimension 3(g − 1). Also, their topological structure is pretty well known; Goulden-Jackson-Harer and Harer-Zagier have found their Euler characteristic, expressing them through the Riemann zeta function. However, topological properties of the whole real locus Rg were less studied. The most known fact is its connectivity, proved independently by Buser-Seppälä-Silhol, Catanese-Frediani and Costa-Izquierdo. This paper can be seen as a further contribution to the study of topology of Rg, which was possible through the notion of the nerve Ng, associated to Rg and called the real nerve. We find upper bounds for its geometrical and homological dimensions and we show their sharpness for infinitely many values of odd g. Precise values of these dimensions for even g have been found by the authors in an earlier paper.


Introduction
By a symmetry of a Riemann surface of genus g we understand its antiholomorphic involution σ.The topological type of the symmetry σ is described by an integer k, whose absolute value is the number of connected components of the set F = Fix(σ) of points fixed by σ, and which is positive if σ is separating, i.e.X − F is disconnected, and negative or 0 otherwise.Each of the components of F is homeomorphic to a circle and called an oval of σ in the Hilbert's nineteenth century terminology.The possible types of individual symmetries are known from the classification of Harnack [14] and Weichold [19].In such a way, the real locus R g of the moduli space M g of compact Riemann surfaces of given genus g can be covered by [(3g + 4)/2] real analytic varieties R k g of dimension 3(g − 1), consisting of the points represented by surfaces having a symmetry of the type k [7,17,18].Also a topological structure of these varieties is well known.In [13,8] the authors found their Euler characteristic expressing them through the Riemann zeta function.
However, topological properties of the whole real locus R g were less studied.The most known fact is its connectivity, proved independently by Buser-Seppälä-Silhol, Catanese-Frediani and Costa-Izquierdo.This paper can be seen as a further contribution to the study of topology of R g , which was possible through the using of the notion of a nerve N g , associated to the covering of R g , by the loci R k g , and called the real nerve.
By the definition, (k 0 , k 1 , . . ., k n ) constitutes an n-simplex of our nerve N g if and only if , which in turn means that there exists a Riemann surface having simultaneously symmetries of distinct types k 0 , k 1 , . . ., k n .The most challenging task here would be to compute the Euler characteristic of N g .This seems, however, to be rather difficult, though calculation of higher-dimensional Betti numbers seems to be more tractable.Here we shall deal with geometrical and homological dimensions of N g .We find upper bounds for them and we show their sharpness for infinitely many values of odd g.Precise values of these dimensions for even g have been found by the authors in an earlier paper [12].
By the mentioned results of of Harnack and Weichold (c.f.[5]), N g has [(3g + 4)/2] points.Moreover, N g is connected by the results of Buser-Seppälä-Silhol [4], Catanese-Frediani and it was also shown by Costa and Izquierdo in [6], that given arbitrary type k of a symmetry of a Riemann surface of genus g, a surface can be chosen in such a way, to also have a symmetry of the type −1.This in fact means that −1 is a spine of N g for any g.
Due to functorial equivalence between compact, connected Riemann surfaces and projective, irreducible, smooth, complex algebraic curves, we can also translate our results to the language of complex curves and their real forms.Under this equivalence, a Riemann surface X admits a symmetry σ if and only if the corresponding curve C X has a real form C X (σ).Moreover, two symmetries σ and τ define real forms C X (σ) and C X (τ ) isomorphic over the reals R if and only if they are conjugate in Aut ± (X).Finally, the set Fix(σ) is homeomorphic to a smooth projective model of the corresponding real form C X (σ).The image M R g , resulting from mapping the moduli space of real algebraic curves into the moduli space M g of complex algebraic curves of genus g, is called the real locus and it is covered by the strata M R g,k , consisting of the points of M g representing complex algebraic curves having a real form, whose smooth projective model has |k| connected components and the set of its R-rational points leaves, when removed, its complexification connected or not according to k being negative (or 0) or positive respectively.

Preliminaries
We obtain our results by using the combinatorial group theory, i.e. theory of noneuclidean crystallographic groups (NEC groups in short), which are the discrete and cocompact subgroups of the group G of all isometries of the hyperbolic plane H.The algebraic structure of such a group Λ is determined by the signature: (1) s(Λ) = (h; ±; [m 1 , . . ., m r ]; {(n 11 , . . ., n 1s 1 ), . . ., (n k1 , . . ., n ks k )}), where the brackets (n i1 , . . ., n is i ) are called the period cycles, the integers n ij are the link periods, m i proper periods and h is the orbit genus of Λ.
A group Λ with signature (1) has the presentation with the following generators, called canonical generators : and relators: is positive, where η = 2 or 1 according to the sign being + or −.This value corresponds to the hyperbolic area µ(Λ) of any fundamental region for Λ and we have the Hurwitz-Riemann formula where Λ ′ is a subgroup of finite index in an NEC group Λ.
NEC groups having no orientation reversing elements are just the classical Fuchsian groups and among them particularly important are the Fuchsian surface groups, which are just torsion free Fuchsian groups.A Fuchsian surface group Γ has signature of the type (g; −) and in such a case H/Γ is a compact Riemann surface of genus g.Conversely, every compact Riemann surface X can be represented as such an orbit space for some Fuchsian surface group Γ and a finite group G is a group of automorphisms of X if and only if G = Λ/Γ for some NEC group Λ.The following result from [9,10] is a main tool that we use in this paper.Theorem 2.1.Let X = H/Γ be a Riemann surface with the group G of all automorphisms of X, let G = Λ/Γ for some NEC group Λ and let θ : Λ → G be the canonical projection.Then the number of ovals of a symmetry σ of X equals where C denotes the centralizer and the sum is taken over a set of representatives of all the conjugacy classes of canonical reflections, whose images under θ are conjugate to σ.
The index ] will be called a contribution of c i to ∥σ∥.The next result of Singerman allows us to compute the centralizer of a canonical reflection in an NEC group Proposition 2.2 ([15, 16]).Let c 0 , c 1 , . . ., c s , e be the canonical generators corresponding to a period cycle (n 1 , . . ., n s ) of an NEC group Λ with signature (1).If all n i are even, then the centralizer A group G is said to be abstractly oriented if there is an epimorphism α : G → Z 2 = {±1}, called an abstract orientation.An element g of abstractly oriented group G with an abstract orientation ε is said to be orientation preserving (respectively orientation reversing) if α(g) = +1 (respectively α(g) = −1).Observe that the abstract orientations of G correspond to the subgroups of index 2 in G.Moreover, by the Sylow theorem, in our studies of symmetries we may assume that G is a 2-group.Indeed, for σ 1 , σ 2 , . . ., σ k being the representatives of conjugacy classes of symmetries we know that all Sylow 2-groups are conjugate and so we can assume that these symmetries generate a 2-group G.In [2] the authors proved the following two theorems: Remark 2.6.Here we shall mention, that one of the possible types of symmetries is 0, which corresponds to a fixed point free (and hence nonseparating) symmetry.As in our studies of the geometrical and homological dimension of the nerve N g we will look for symmetries with distinct types, we have to take a fixed point free symmetry into account as one (and only one!) of the possibilities.It was shown in [3] that the bound from corollary 2.5 is also true, if we allow fixed point free symmetries.
For the sake of completeness, before we move to the main sections of the paper, let us cite results concerning even values of g, which were obtained in [12].

Geometrical dimension of N g
Since symmetries of a Riemann surface X having distinct topological types are nonconjugate in Aut ± (X), the quantitative results concerning symmetries from the corollary 2.5 give us upper bounds for dim G (N g ) and dim H (N g ).Observe, however, that to study the attainment, one needs qualitative results allowing topological type of single symmetry σ of X to be found in terms of Aut ± (X) and topological type of the action.This role in our paper will be played by theorem 2.1.Here we shall prove the following Theorem 3.1.Let g = 2 r−1 u + 1 where r ≥ 2 and u is odd.The geometrical dimension of N g does not exceed 2 r+1 − 1 and this bound is attained if u ≥ 2 r (2 r + 1) − 5.
Example 3.2.Before we present the proof of the above theorem, let us consider an example, which shall give us an idea and intuition for the upcoming proof.The example can be viewed as the proof for the specific case r = 2 in the theorem above.
Let g = 2u + 1 for some odd integer u ≥ 15.It is obvious, by corollary 2.5, that the geometrical dimension of N g cannot be greater than 7. Indeed, there are at most 8 nonconjugate symmetries, hence at most 8 symmetries of distinct types and hence the dimension of a simplex in N g cannot be greater than 7.We shall show that in fact it is maximal and equals 7 in such a case.Take G = Z 4  2 to be an abstractly oriented group having 8 orientation reversing involutions σ 0 , σ 1 , . . ., σ 7 .Consider an NEC group with signature (0; +; [2, (u−15)/2 . . ., 2]; {(2, 19  . .., 2)}), and define and epimorphism θ : Λ → Z 4 2 on a sequence of consecutive canonical reflections in the following way: Moreover, we define θ(x l ) = σ 0 σ 1 for all values of l and θ(e 1 ) = 1 if (u − 15)/2 is even and θ(e 1 ) = σ 0 σ 1 otherwise.Here we consider our reflections as situated on a circle, so we unify the first and the last reflection.It is easy to see that symmetry σ i has 2i ovals.Indeed, whenever a symmetry appears with the same neighbors, the corresponding reflection contributes to it with 4 ovals, by theorem 2.1 and proposition 2.2.Now if it appears with distinct neighbors we have a contribution of 2 ovals to the respective symmetry.It follows easily, that we constructed a Riemann surface H/ ker θ, having 8 nonconjugate symmetries, each with different number of ovals, hence of distinct topological types.Therefore we constructed a 7-dimensional simplex and the geometrical dimension here is maximal and equals 7.
Proof.The upper bound on dim G N g is obvious by corollary 2.5.Again, as there are at most 2 r+1 nonconjugate symmetries, then there are at most 2 r+1 distinct types.It follows immediately that the dimension of a simplex in N g is bounded by 2 r+1 − 1 and so dim G (N g ) ≤ 2 r+1 −1.Therefore for the proof it is enough to construct, for any g as in the theorem, a Riemann surface of genus g, having 2 r+1 symmetries of distinct topological types.Let u ≥ 2 r (2 r + 1) − 5 and consider an NEC group Λ with signature where G is an abstractly oriented group.Denote by σ i , 0 ≤ i ≤ 2 r+1 −1, all the orientation reversing involutions in G. Let θ(x l ) = σ 0 σ 1 for 1 ≤ l ≤ m and θ(e 1 ) = 1 or σ 0 σ 1 depending on the parity of m, that is θ(e 1 ) = 1 for m being even and θ(e 1 ) = σ 0 σ 1 otherwise.Let us divide the sequence of canonical reflections c 0 , . . ., c s−1 in the following way: first we have 2 r segments, where the n-th segment has length 2 r+1 − (2n − 1), and at the end we have the remaining 2 r − 1 reflections; we shall call these a tail of the sequence.Now we define θ to map the reflections of n-th segment alternatively to σ 2 r+1 −(2n−1) and σ 2 r+1 −2n , starting and finishing with σ 2 r+1 −(2n−1) .The last 2 r -th segment has length 1 and its reflection is mapped to σ 1 .Next, we map the remaining consecutive reflections respectively to As before, we unify the first and the last reflection, hence we can view the canonical reflections as situated on a circle and treat the first one and the last one as the same.Clearly θ is an epimorphism and it is also easy to determine the number of ovals of all the symmetries.Obviously σ 0 is a fixed point free symmetry.The centralizer of any symmetry σ i in G has order 2 r+2 .By proposition 2.2, an image of the centralizer of a reflection c in Λ is generated by θ(c) = σ i and the images of its neighboring reflections on a circle mentioned above.Therefore, if these images are distinct, then the reflection c contributes, by theorem 2.1, with 2 r+2 /8 = 2 r−1 ovals to symmetry σ i .If these images are the same, then we have 2 r+2 /4 = 2 r ovals.
Let us now consider symmetries of the n-th segment, σ 2 r+1 −(2n−1) and σ 2 r+1 −2n for some 1 ≤ n ≤ 2 r .The segment has odd length and its first and last reflections contribute with 2 r−1 ovals to symmetry σ 2 r+1 −(2n−1) .Indeed, recall that for the first and last reflection in the segment, the images of neighboring reflections are distinct, so the image of this centralizer has order 8. Now by the theorem 2.1, the last and the first reflection in the segment contribute |G|/8 = 2 r−1 ovals each.Observe also, that all the remaining reflections of this segment contribute with |G|/4 = 2 r ovals to respective symmetries.This is so, because now the image of the centralizer of a reflection has order 4, as the neighboring reflections have the same image.Summing up, the n-th segment gives (2 r − n) • 2 r ovals to symmetries σ 2 r+1 −(2n−1) and σ 2 r+1 −2n each.Furthermore, symmetry σ 2 r+1 −(2n−1) appears once more, as it is the image of one of the reflections at the tail of the cycle (not connected to any segment).This reflection again contributes with 2 r−1 ovals.Therefore, symmetry Observe also that symmetries connected with distinct segments have distinct numbers of ovals, as the lengths of the segments differ.Hence we arrived to the configuration of 2 r+1 symmetries σ i , where the symmetry σ i has i • 2 r−1 ovals, for some 0 ≤ i ≤ 2 r+1 − 1.This shows that we have a (2 r+1 − 1)-simplex in N g and so dim G N g = 2 r+1 − 1, the proof is finished

Homological dimension of N g
Now we shall deal with the problem of the homological dimension dim H N g .Obviously the homological dimension of the nerve cannot be greater than the geometrical dimension of N g .The next result shows, that in fact the bound again is attained for infinitely many values of g.Theorem 4.1.Let g = 2 r−1 u + 1 for some r ≥ 2 and u odd.Then the homological dimension of N g does not exceed 2 r+1 − 1 and equals

Example 4.2. The first part of the statement is clearly true, as dim
As before, we shall start with a specific case of r = 2, which shall help us to go through the general proof.Let us assume that g = 2u + 1 for some odd u ≥ 19.We shall construct 9 Riemann surfaces X 0 , . . ., X 8 in such a way, that a surface X j has 8 commuting symmetries σ i with 2i ovals each, where 0 ≤ i ≤ 8 and i ̸ = j.Let us take an NEC group Λ with signature (2).Now we shall consider 9 cases defining m, s and respective epimorphisms θ j onto Z 4  2 such that X j = H/ ker θ j is the surface we looked for.Throughout the proof we assume σ i to be all the orientation reversing involutions in Z 4  2 .We also assume that the canonical reflections in an NEC group Λ are situated on a circle so that the last and first one are unified and treated as one.and all the elliptic generators to σ 1 σ 2 .The connecting generator e 1 is mapped to σ 1 σ 2 or 1 for m odd or even respectively.By theorem 2.1 and proposition 2.2, in the same way as we did in the proof of theorem 3.1, it is easy to see that σ i has 2i ovals for i = 1, . . ., 8.Here again σ i has 2i ovals for i = 2, . . ., 7 and i = 0. Observe also that in this case we have in addition a fixed point free symmetry σ 0 .This also holds true for all the remaining cases.The elliptic and connecting generators are mapped as in the previous case.The elliptic and connecting generators are mapped as in the previous case.
The elliptic and connecting generators are mapped as in the previous case.
Case 8: Take m = u−15 2 , s = 19 and define θ 8 by mapping the consecutive canonical reflections as in the first Example concerning theorem 3.1.It is not hard to see, that in the j-th Case we obtained a surface X j with the configuration of symmetries announced before.
Proof.Let g be as in the theorem with u ≥ 2 r+1 (2 r−1 + 1) − 5. Our aim will be to construct 2 r+1 + 1 Riemann surfaces X j , j = 0, 1, . . ., 2 r+1 of genus g, each having 2 r+1 symmetries with 0 • 2 r−1 , 1 • 2 r−1 , . . ., j • 2 r−1 , . . ., 2 r+1 • 2 r−1 ovals, where the symbol • means that the corresponding value is removed.We shall divide our considerations into a few cases, depending on the 4-adic structure of j − 1.Throughout this part of the proof we shall again denote G = Z r+2 2 .We also employ the following convention: during the construction of X j assume that all the orientation reversing involutions in G are σ i for i ̸ = j and 0 ≤ i ≤ 2 r+1 .These will also become symmetries in question and the convention will allow us to link the number of the symmetry with its number of ovals, which for symmetry σ i will be equal to i • 2 r−1 .In addition, during constructions of the epimorphisms θ j , j ̸ = 0, j ̸ = 1 for which X j = H/ ker θ j , we assume that all the m elliptic generators are mapped to σ 0 σ 1 and θ j (e 1 ) = σ 0 σ 1 for m odd and θ j (e 1 ) = 1 otherwise.For j = 0 we replace σ 0 σ 1 in the above definition with σ 1 σ 2 , and for j = 1 we take it to be σ 0 σ 2 .
First of all we shall construct X 0 , which will be a Riemann surface having 2 r+1 symmetries respectively with i • 2 r−1 ovals, where 1 ≤ i ≤ 2 r+1 .Consider an NEC group Λ with signature (2), where m = (u − 2 r+1 (2 r−1 + 1) + 5)/2 and s = 2 r+1 (2 r−1 + 1) − 1.Let σ 1 , . . ., σ 2 r+1 denote all the symmetries in G.We define θ 0 : Λ → G in the following way: we divide our cycle into pieces such that first we have 2 r − 1 segments, where the n-th segment has length 2 r+1 − 2(n − 1).The consecutive canonical reflections corresponding to the n-th segment are sent alternatively to σ 2 r+1 −2(n−1) and σ 2 r+1 −2n+1 , starting with the former and finishing with the latter.The last of these segments has length 4 and its reflections are sent to σ 4 , σ 3 , σ 4 , σ 3 .Now the next reflection is mapped to σ 1 and the remaining reflections respectively to σ 4 , σ 6 , . . ., σ 2 r+1 , σ 2 , as shown below: In the same way as in the proof of theorem 3.1, we see that the reflections of the n-th segment are contributed with (2 r+1 − 2n + 1) • 2 r−1 ovals each.Furthermore, symmetry σ 2 r+1 −2(n−1) appears once again as the image of one of the reflections at the end of the cycle.Therefore it has (2 r+1 − 2n + 2) • 2 r−1 ovals.In addition, symmetry σ 1 has 2 r−1 ovals and symmetry σ 2 has 2 r ovals.Hence the epimorphism θ 0 leads to the configuration of 2 r+1 symmetries σ i with i • 2 r−1 ovals each, where 1 ≤ i ≤ 2 r+1 .We construct the surface X 2 in the similar way.Consider an epimorphism θ 2 : Λ → G, which is defined in the same way as θ 0 on all the canonical reflections except the last segment and the tail of the cycle.On these canonical reflections we define θ 2 as Compared to θ 0 , we do not have the symmetry with 2 • 2 r−1 ovals, which was our aim.This one is replaced by fixed point free symmetry σ 0 .The numbers of ovals of all the other symmetries did not change.Note, that we replaced the symmetry σ 2 at the end of the cycle with symmetry σ 4 but we also changed the sequence of the symmetries at the end of the cycle and switched σ 3 with σ 1 in the last segment.Therefore also now σ 4 has 4 • 2 r−1 ovals, although it appears once more in the cycle.Now we shall construct surfaces X 2n for 1 < n ≤ 2 r .Consider an NEC group Λ with signature (2) where m = (u − 2 r+1 (2 r−1 + 1) + 5)/2 + ⌊n/2⌋ and s = 2 r+1 (2 r−1 + 1) − 1 − 2⌊n/2⌋.We define θ 2n : Λ → G by dividing our cycle into pieces, as before, such that first we have 2 r − 1 segments and after these segments we have the tail of the cycle.Now the segments with numbers from 1 to 2 r − n are the same as in the case of X 0 and we define θ 2n in the same way as θ 0 on the canonical reflections corresponding to these segments.Now we shall modify the latter part, depending on the parity of n.
Let first n be even.The next segments (with numbers from 2 r − n + 1 up to 2 r − 1) are shortened and their lengths are diminished by 1, which means they have lengths 2n − 1, 2n − 3, . . ., 3 respectively.The consecutive canonical reflections corresponding to these shortened segments are mapped as follows: Hence for the a-th segment (with a ≥ 2 r − n + 1) the reflections are sent alternatively to σ 2 r+1 −2a+1 and σ 2 r+1 −2a , starting and finishing with the former.These segments are exactly the same as in the proof of theorem 3.1.Now the consecutive reflections from the tail of the cycle, not belonging to the segments, are mapped respectively to σ 2 r+1 , σ 2 r+1 −2 , . . ., σ 2n+2 , σ 2n−1 , σ 2n−3 , . . ., σ 1 .Obviously the numbers of ovals of symmetries with numbers distinct from 2n have not changed compared to θ 0 .But here we do not have symmetry with 2n • 2 r−1 ovals and in fact this one has been replaced by fixed point free symmetry σ 0 , which is the configuration we looked for.
Let now n be odd.The only difference we make, compared to the case of n being even, is on the last segment and the latter part of the cycle.Recall that the last segment had length 3 and its reflections were mapped to σ 3 , σ 2 , σ 3 .We take this segment to have length 4 and map the corresponding reflections to σ 3 , σ 1 , σ 2 , σ 3 .Observe that this operation causes the symmetry σ 2 to loose 2 r−1 ovals.But we also modify the epimorphism on the last part of the cycle, by taking the reflections to be mapped to Here the total length of the cycle remains correct and again there is no symmetry σ 2n with 2n • 2 r−1 ovals, this symmetry was replaced by σ 0 .As for the other symmetries, the only difference is that the symmetry σ 2 appeared once more at the end of the cycle, which gave her the lacking 2 r−1 ovals -the ones that were taken during the modification of the last segment.Now we shall construct surfaces X 4n+3 for 0 ≤ n < 2 r−1 − 1.Consider an NEC group Λ with signature (2) where m = (u − 2 r+1 (2 r−1 + 1) + 5)/2 + (n + 1) and s = 2 r+1 (2 r−1 + 1) − 1 − 2(n + 1).We define θ 4n+3 : Λ → G in the similar way as in the previous case.Like before, we divide our cycle into pieces such that first we have 2 r − 1 segments and after these segments we have the tail of the cycle.On the first segments, up to the number 2 r − 2n − 2, we define the epimorphism as in the case of θ 0 , which means that a-th segment has length 2 r+1 − 2(a − 1) and its reflections are sent alternatively to σ 2 r+1 −2(a−1) and σ 2 r+1 −2a+1 , starting with the former and finishing with the latter.We change the epimorphism on the next segment, with number 2 r −2n−1, by replacing all the entries of symmetry σ 4n+3 by symmetries σ 2n+2 and σ 2n+1 .As a result, the epimorphism θ 4n+3 sends the reflections of this segment respectively to: Observe that in fact the symmetries σ 2n+2 , σ 2n+1 cannot appear again in the sequence of images of the canonical reflections.Moreover, the symmetry σ 4n+4 lost 2 r−1 ovals.Observe also that for n = 0 this modified segment is the last one and it consists of σ 4 , σ 2 , σ 4 , σ 1 , while in the other cases the last segment consists of two reflections which are mapped to σ 2 , σ 1 .On the segments with numbers from 2 r − 2n up to 2 r − n − 1, if there are any, again we define the epimorphism as in the case of θ 0 : the a-th segment has length 2 r+1 − 2(a − 1) and its reflections are sent alternatively to σ 2 r+1 −2(a−1) and σ 2 r+1 −2a+1 , starting with the former and finishing with the latter.Now for the segment with number 2 r − n we should have symmetries σ 2n+2 , σ 2n+1 , but these were already used before.Therefore we skip these symmetries and take the next pair, if there is any.As a result, all the segments with numbers from 2 r − n up to 2 r − 1 are, roughly speaking, 'shifted' compared to what we had with θ 0 and a-th segment has length 2 r+1 − 2a, its reflections being sent alternatively to σ 2 r+1 −2a and σ 2 r+1 −2a−1 , starting with the former and finishing with the latter.The last thing here is to define the epimorphism on the tail of the cycle and we do this by mapping the consecutive reflections respectively to σ 2 r+1 −2 , σ 2 r+1 −4 , . . ., σ 4n+6 , σ 4n+4 , . . ., σ 2n+4 , σ 2n+2 , σ 2n , . . ., σ 2 , σ 2 r+1 , σ 4(n+1) for 0 ≤ n ≤ 2 r−1 − 2 (note that for n = 0 we have σ 4n+4 = σ 2n+4 and the last appears only at the end of the cycle and σ 2 does not appear at all).Here it is easy to see that we obtained a configuration we were looking for, that is a set of 2 r+1 symmetries, on a Riemann surface of genus g, where the symmetries have i • 2 r−1 ovals for 0 ≤ i ≤ 2 r+1 and i ̸ = 4n + 3 for some 0 ≤ n < 2 r−1 − 1.Now if n = 2 r−1 − 1, we use the same definitions for Λ and in the epimorphism we only change the last segment and the tail of the sequence.That is, we take the last segment, with number 2 r − 1, to be σ 1 , σ 2 and the tail of the sequence to be: With these definitions we obtain a desired configuration of symmetries.Now we shall construct X 4n+1 for 1 ≤ n ≤ 2 r−1 − 1.Consider an NEC group Λ with signature (2), where m = (u−2 r+1 (2 r−1 +1)+5)/2+(n+1), s = 2 r+1 (2 r−1 +1)−1−2(n+1) We define θ 4n+1 : Λ → G in the following way: we divide our cycle into pieces such that first we have 2 r − 2 segments and after these segments we have the tail of the cycle.On the first segments, up to the number 2 r − 2n − 1, we define the epimorphism as in the case of θ 0 .We change the next segment, with number 2 r − 2n and length 4n + 2, by changing its length to 4n + 4 and the epimorphism on this segment by replacing symmetry σ 4n+1 by symmetries σ 2n+2 and σ 2n+1 .As a result, the epimorphism θ 4n+1 sends the reflections of this segment respectively to: Again symmetries σ 2n+2 , σ 2n+1 already have (2n + 2) • 2 r−1 and (2n + 1) • 2 r−1 ovals and will not appear again in the sequence of images of the canonical reflections.Moreover, the symmetry σ 4n+2 gained 2 r−1 ovals and also will not appear again.Now for the segments with numbers from 2 r −2n+1 up to 2 r −n−1, if any exist, again we define the epimorphism as in the case of θ 0 .Similarly, for the segment with number 2 r − n, if it exists, we should have symmetries σ 2n+2 , σ 2n+1 , but as these were already used before, we take the next pair.As a result, all the segments (if there are any) with numbers from 2 r − n up to 2 r − 2 are 'shifted' compared to what we had with θ 0 and a-th segment has length 2 r+1 − 2a, its reflections being sent alternatively to σ 2 r+1 −2a and σ 2 r+1 −2a−1 , starting with the former and finishing with the latter.The last thing here is to define the epimorphism on the last part of the cycle and we do this by mapping the consecutive reflections respectively to σ 1 , σ 4 , σ 6 , . . ., σ 2n+2 , σ 2n+4 , . . ., σ 4n+2 , σ 4n+4 , . . ., σ 2 r+1 , σ 2 for 1 ≤ n ≤ 2 r−1 − 1.In this sequence any symmetry appears only once and we remove σ 2n+2 and σ 4n+2 .Note that for n = 1 we have 2n + 2 = 4, 4n + 2 = 6 and so σ 4 and σ 6 do not appear at all in the sequence above.This again leads to the configuration we were looking for.
The last thing is to construct X 1 .Here again we take an NEC group with signature (2), where m = (u − 2 r+1 (2 r−1 + 1) + 5)/2 and s = 2 r+1 (2 r−1 + 1) − 1.The epimorphism θ 1 : Λ → G is defined in the same way as above but we need to change the tail of the sequence slightly, as the symmetry removed is σ 1 .We start with 2 r − 1 segments, where the a-th segment has length 2 r+1 − 2(a − 1) and maps the consecutive reflections alternatively to σ 2 r+1 −2(a−1) and σ 2 r+1 −2(a−1)−1 , starting with the former and finishing with the latter.The last segment has length 4 and its symmetries are σ 4 , σ 3 , σ 4 , σ 3 .Now for the tail of the sequence we take: This construction gives a Riemann surface X 1 , having 2 r+1 nonconjugate symmetries of distinct topological types, as the symmetries have i • 2 r−1 ovals for i ̸ = 1 and 0 ≤ i ≤ 2 r+1 .Remark 4.3.In the two main theorems of this paper we constructed surfaces having the desired configuration of symmetries with distinct numbers of ovals and hence with distinct types.However, as in our constructions we used the abelian group G as the automorphism group of the surface, by the results of [1] it is not hard to determine that all the symmetries constructed are nonseparating.

Concluding remarks, open problems and conjectures
We would like to finish this paper with some conjectures, which are being investigated right now and as far as we are concerned at the moment, they are most probably true.
The first conjecture concerns the structure of the group generated by the so-called extremal configuration of symmetries.This basically means, by theorem 2.4, that given a Riemann surface of genus g = 2 r−1 u + 1, for some odd integer u and such that it has 2 r+1 nonconjugate symmetries, the structure of the group generated by the symmetries is just a direct product of a dihedral group and respective amount of cyclic groups of order 2.This result can be seen as the generalization of one of the theorems of Gromadzki and Izquierdo from [11].Being given the structure of the group generated by symmetries in question, we should have enough data to prove the next conjecture to be true.Conjecture 5.2.The sufficient conditions for the maximal geometrical and homological dimensions of N g , given in theorems 3.1 and 4.1, are also necessary.
We claim that actually the lower bounds on integer u, and hence the lower bounds on g cannot be improved.As for the geometrical dimension, it seems that actually it is enough to consider an abelian group G, being a direct product Z r+2 2 and an NEC group signature (2).The important thing here is that when we get rid of the non-zero genus and multiple period cycles in the signature of Λ, we are left with consecutive reflections situated on a circle.This basically means, that we only have to solve the following combinatorial problem for k = 2 r+1 − 1.
Problem 5.3.Let us consider a number of points situated on a circle, coloured by k ≥ 3 colours in such a way that no two consecutive points have the same colour.Moreover, we put weights on our points in such a way that the weight is 2 if a point has neighbours with the same colour and the weight is 1 otherwise.Next, for every colour we define its weight as the sum of all the weights of points coloured with it.What is the smallest possible number of points φ(k), for which there exists such a colouring and all the colours have distinct weights?For our principal goals, the most important is the case k = 2 r+1 − 1 for which we have some evidence for the following conjecture to be true.The point is that this problem, k = 2 r+1 − 1, describes exactly the situation, when we need to have k = 2 r+1 − 1 symmetries with fixed points (colours), in addition one fixed point free symmetry, and the group epimorphism can be seen as the colouring of the points (consecutive canonical reflections).Looking for the smallest possible number of points is just looking for the shortest possible period cycle, hence the group Λ with smallest area and hence, by the Hurwitz-Riemann formula, the smallest possible g for which there exists a Riemann surface X of genus g, realizing the maximal geometrical dimension of the nerve N g .We also believe that it will be possible to employ this method to solve similar, but more difficult, problem for the homological dimension of N g .Finally, given k and s ≥ φ(k), a systematic procedure of producing all colourings, described in Problem 5.3, is also crucial to find higher Betti numbers of N g .

Conjecture 5 . 1 .
Let G be an abstractly oriented 2-group generated by orientation reversing involutions.If G contains a dihedral group D n as a subgroup of index 2 r , an element of order n in D n preserves the orientation and G has 2 r+1 conjugacy classes of orientation reversing involutions, then it is a direct product D n ⊕ Z 2 ⊕ r . . .⊕Z 2 .

Theorem 2.3. A 2-group G containing a dihedral group D n as a subgroup of index 2 r has at most 2 r+2 − 1 conjugacy classes of elements of order 2. Furthermore, if G is abstractly oriented and the generators x 0 , y 0 of D n reverse the orientation, then G has at most 2 r+1
conjugacy classes of orientation reversing elements of order 2.