Symmetric Genus Research Articles

The symmetric genus spectrum of abelian groups

Let S denote the set of positive integers that appear as the symmetric genus of a finite abelian group and let S 0 denote the set of positive integers that appear as the strong symmetric genus of a finite abelian group. The main theorem of this paper is that S = S 0 . As a result, we obtain a set of necessary and sufficient conditions for an integer g to belong to S . This also shows that S has an asymptotic density and that it is approximately 0.3284 .

Regular dessins d'enfants with dicyclic group of automorphisms

Let Gn be the dicyclic group of order 4n. We observe that, up to isomorphisms, (i) for n≥2 even there is exactly one regular dessin d'enfant with automorphism group Gn, and (ii) for n≥3 odd there are exactly two of them. Each of them is produced on well known hyperelliptic Riemann surfaces. We obtain that the minimal genus over which Gn acts purely-non-free is σp(Gn)=n (this coincides with the strong symmetric genus of Gn when n is even). For each of the triangular conformal actions, every non-trivial subgroup of Gn has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of Gn, of its jacobian variety has only one component. We also study conformal/anticonformal actions of Gn, on closed Riemann surfaces, with the property that Gn admits anticonformal elements. It is known that Gn always acts on a genus one Riemann surface with such a property. We observe that the next genus σhyp(Gn)≥2 over which Gn acts in that way is n+1 for n≥2 even, and 2n−2 for n≥3 odd. We also provide examples of pseudo-real Riemann surfaces admitting Gn as the full group of conformal/anticonformal automorphisms.

The strong symmetric genus spectrum of nilpotent groups

Let denote the set of positive integers that appear as the strong symmetric genus of a finite nilpotent group. We show that if g is not congruent to , then . Using this result and some abelian groups with genus congruent to , we establish a lower bound of 8/9 for the lower asymptotic density of the set . We also prove that there are an infinite number of positive integers not in the set .

Automorphism groups of dessins d’enfants

Recently, Gareth Jones observed that every finite group $G$ can be realized as the group of automorphisms of some dessin d'enfant ${\mathcal D}$. In this paper, complementing Gareth's result, we prove that for every possible action of $G$ as a group of orientation-preserving homeomorphisms on a closed orientable surface of genus $g \geq 2$, there is a dessin d'enfant ${\mathcal D}$ admitting $G$ as its group of automorphisms and realizing the given topological action. In particular, this asserts that the strong symmetric genus of $G$ is also the minimum genus action for it to acts as the group of automorphisms of a dessin d'enfant of genus at least two.

Geometry of the Wiman\u2013Edge pencil, I: algebro-geometric aspects

In 1981 William L. Edge discovered and studied a pencil mathscr {C} of highly symmetric genus 6 projective curves with remarkable properties. Edge’s work was based on an 1895 paper of Anders Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel Geometry of the Wiman–Edge pencil, II: hyperbolic, conformal and modular aspects (in preparation), we consider mathscr {C} from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.

The strong symmetric genus spectrum of abelian groups

Let $${\mathcal {S}}$$ denote the set of positive integers that may appear as the strong symmetric genus of a finite abelian group. We obtain a set of (simple) necessary and sufficient conditions for an integer g to belong to $${\mathcal {S}}$$ . We also prove that the set $${\mathcal {S}}$$ has an asymptotic density and approximate its value.

The density of the strong symmetric genus values of p-groups

ABSTRACTLet G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Let p a prime, and let Jp be the set of integers g for which there is a p-group of strong symmetric genus g. We show that the set Jp has density zero in the set of positive integers.

On the Symmetric Genus of K-metacyclic Group and Its Presentations

On the Symmetric Genus of K-metacyclic Group and Its Presentations

Genus of the Cartesian Product of Triangles.

We investigate the orientable genus of $G_n$, the cartesian product of $n$ triangles, with a particular attention paid to the two smallest unsolved cases $n=4$ and $5$. Using a lifting method we present a general construction of a low-genus embedding of $G_n$ using a low-genus embedding of $G_{n-1}$. Combining this method with a computer search and a careful analysis of face structure we show that $30\le \gamma(G_4) \le 37$ and $133 \le\gamma(G_5) \le 190$. Moreover, our computer search resulted in more than $1300$ non-isomorphic minimum-genus embeddings of $G_3$. We also introduce genus range of a group and (strong) symmetric genus range of a Cayley graph and of a group. The (strong) symmetric genus range of irredundant Cayley graphs of $Z_p^n$ is calculated for all odd primes $p$.

The Symmetric Genus of p-Groups

Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has symmetric genus not congruent to 1(mod p 3) if and only if G is in one of 10 families of groups. The genus formula for each of these 10 families of groups is determined. A consequence of this classification is that almost all positive integers that are the genus of a p-group are congruent to 1(mod p 3). Finally, the integers that occur as the symmetric genus of a p-group with Frattini-class 2 have density zero in the positive integers.

THE SYMMETRIC GENUS OF 2-GROUPS

AbstractLet G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2m−3 and a dihedral subgroup of index 4 or else the exponent of G is 2m−2. We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

THE STRONG SYMMETRIC GENUS OF DIRECT PRODUCTS

Let G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Assume that G is non-abelian and generated by two elements, one of which is an involution, and that n is relatively prime to |G|. Our first main result is the determination of the strong symmetric genus of the direct product Zn ×G in terms of n, |G|, and a parameter associated with the group G. We obtain a variety of genus formulas. Finally, we apply these results to prove that for each integer g ≥ 2, there are at least four groups of strong symmetric genus g.

The symmetric genus spectrum of finite groups

The symmetric genus of the finite group G , denoted by σ ( G ), is the smallest non-negative integer g such that the group G acts faithfully on a closed orientable surface of genus g (not necessarily preserving orientation). This paper investigates the question of whether for every non-negative integer g , there exists some G with symmetric genus g . It is shown that that the spectrum (range of values) of σ includes every non-negative integer g =!= 8 or 14 mod 18, and moreover, if a gap occurs at some g == 8 or 14 modulo 18, then the prime-power factorization of g − 1 includes some factor p e == 5 mod 6. In fact, evidence suggests that this spectrum has no gaps at all.

The action of the groups D m × D n on unbordered Klein surfaces

Every finite group G may act as an automorphism group of Klein surfaces either bordered or unbordered either orientable or non-orientable. For each group the minimum genus receives different names according to the topological features of the surface X on which it acts. If X is a bordered surface the genus is called the real genus ρ(G). If X is a non-orientable unbordered surface the genus is called the symmetric crosscap number of G and it is denoted by \({\widetilde{\sigma}(G)}\). Finally if X is a Riemann surface it has two related parameters. If G only contains orientation-preserving automorphisms we have the strong symmetric genus, σ0(G). If we allow orientation-reversing automorphisms we have the symmetric genus σ(G). In this work we obtain the strong symmetric genus and the symmetric crosscap number of the groups Dm × Dn. The symmetric genus of these groups is 1. However we introduce and obtain a new parameter, denoted by τ as the least genus g ≥ 2 of Riemann surfaces on which these groups act disregarding orientation.

THE 2-GROUPS OF ODD STRONG SYMMETRIC GENUS

Let G be a finite group. The strong symmetric genusσ0(G) is the minimum genus of any Riemann surface on which G acts preserving orientation. We show that a 2-group G has strong symmetric genus congruent to 3 (mod 4) if and only if G is in one of 14 families of groups. A consequence of this classification is that almost all positive integers that are the genus of a 2-group are congruent to 1 (mod 4).

ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4

AbstractUsing uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

The strong symmetric genus and generalized symmetric groups G(n, 3)

The generalized symmetric groups are defined to be the groups G(n, m) = ℤm ≀ Σm where n, m ∈ ℤ+. The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation-preserving automorphisms. The present paper extends work on the strong symmetric genus by Conder, who studied the symmetric groups, which are the groups G(n, 1), and the author, who studied the hyperoctahedral groups, which are the groups G(n, 2). We determine the strong symmetric genus of the groups G(n, 3).

The symmetric cross-cap number of the groups Cm × Dn

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric cross-cap number and denoted by $\tilde{\sigma}(G)$. This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus.The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm × Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.

The strong symmetric genus of the finite Coxeter groups

The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. In this paper we complete the calculation of the strong symmetric genus for each finite Coxeter group excluding the group E8.