Algebraic Genus Research Articles

Linear representations of fundamental groups of Klein surfaces derived from spinor representations of Clifford algebras

We study actions of multiplicative subgroups of Clifford algebras on Riemann surfaces. We show that every Klein surface of algebraic genus greater than 1 is isomorphic to the orbit space of such an action. We obtain linear representations of fundamental groups of Klein surfaces by using the spinor representations of Clifford algebras.

Full automorphism groups of large order of compact bordered Klein surfaces

Let S be a compact bordered Klein surface of algebraic genus g≥2, and Aut(S) its full group of automorphisms, which is known to have order at most 12(g−1). In this paper we consider groups G of automorphisms of order at least 4(g−1) acting on such surfaces, and study whether G is the full group Aut(S) or, on the contrary, the action of G extends to a larger group. The extendability of the action depends first on the NEC signature with which G acts and, in some cases, also on whether a monodromy presentation of G admits or not a particular automorphism. For each signature we study which of the three possibilities [Aut(S):G]=1, 2 or 3 occur, and show that, whenever a possibility occurs, it occurs for infinitely many values of g. We find infinite families of groups G, explicitly described by generators and relations, which satisfy the corresponding equality.

On the elliptic null-phase solutions of the Kulish–Sklyanin model

We consider Kulish–Sklyanin model (KSM) which is a three-component nonlinear Schrödinger system. Using Dubrovin’s method we derive recurrent relations which allow us to derive all equations in the relevant hierarchy. Thus we derive the first two equations of the Kulish–Sklyanin hierarchy and use them to construct multi-phase solutions of KSM. To do this we first express the monodromy matrix M of the Lax operator L in terms of an elliptic function v(t). In order to derive the solutions of the hierarchies we calculate the corresponding Wronskians which allow us to find also the spectral curves. An important role here plays the matrix Jn in M. Here we have two possibilities: Jn can be either generic block-diagonal or simply diagonal matrix. Thus we are able to derive two classes of solutions of the KSM and demonstrate principal differences between KSM and the other well known integrable models. The first one is that the multi-phase solutions of KSM are more involved than the corresponding solutions of the other hierarchies. The second one is that the algebraic genus of the spectral curves does not always coincide with the number of the phases.

On classical upper bounds for slice genera

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

Lifting the hyperelliptic involution of a Klein surface

We consider unbranched normal coverings $$X\rightarrow X'$$ between compact Klein surfaces of algebraic genus bigger than one where $$X'$$ is hyperelliptic. Here unbranched means that the fixed point set of the group G of covering transformations is either empty or projects onto the boundary of $$X'$$ . We find a criterion which determines whether the hyperelliptic involution of $$X'$$ lifts to an automorphism of X. The study splits naturally into six cases according to the different topological types that $$X'$$ may possess. Our results apply nicely to the case when G is abelian, showing for instance that if G has odd order then the hyperelliptic involution always lifts. We also apply the criterion to particular presentations of other types of groups.

Bordered surfaces in the 3-sphere with maximum symmetry

We consider orientation-preserving actions of finite groups G on pairs (S3,Σ), where Σ denotes a compact connected surface embedded in S3. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus g>1 the maximum order of such a G for all embeddings of a surface of genus g, and classified the corresponding embeddings.In the present paper we obtain analogous results for the case of bordered surfaces Σ (i.e. with non-empty boundary, orientable or not). Now the genus g gets replaced by the algebraic genus α of Σ (the rank of its free fundamental group); for each α>1 we determine the maximum order mα of an action of G, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into S3. For example, the maximal possibility 12(α−1) is obtained for the finitely many values α=2,3,4,5,9,11,25,97,121 and 241.

On Calculating the Slice Genera of 11- and 12-Crossing Knots

ABSTRACTThis article contains the results of efforts to determine the values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus, further upper bounds were produced using the algebraic genus. Lower bounds were obtained using a new obstruction from the Seifert form and by the use of Donaldson’s diagonalization theorem. These results complete the computation of the topological slice genera for all knots with at most 11 crossings and leaves the smooth genera unknown for only two 11-crossing knots. For 12 crossings, there remain merely 25 knots whose smooth or topological slice genus is unknown.

ON TOPOLOGICAL CLASSIFICATION OF FINITE CYCLIC ACTIONS ON BORDERED SURFACES

In Hirose (Tohoku Math. J. 62 (2010), 45–53), Susumu Hirose showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_{g}$ of genus $g\geqslant 2$ determines the group up to a topological conjugation, provided that $N\geqslant 3g$. Gromadzki et al. undertook in Bagiński et al. (Collect. Math. 67 (2016), 415–429) a more general problem of topological classification of such group actions for $N>2(g-1)$. In Gromadzki and Szepietowski (Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110 (2016), 303–320), we considered the analogous problem for closed nonorientable surfaces, and in Gromadzki et al. (Pure Appl. Algebra 220 (2016), 465–481) – the problem of classification of cyclic actions generated by an orientation-reversing self-homeomorphism. The present paper, in which we deal with topological classification of actions on bordered surfaces of finite cyclic groups of order $N>p-1$, where $p$ is the algebraic genus of the surface, completes our project of topological classification of ‘‘large” cyclic actions on compact surfaces. We apply obtained results to solve the problem of uniqueness of the actions realizing the solutions of the so-called minimum genus and maximum order problems for bordered surfaces found in Bujalance et al. (Automorphisms Groups of Compact Bordered Klein Surfaces: A Combinatorial Approach, Lecture Notes in Mathematics 1439, Springer, 1990).

Abelian actions on compact bordered Klein surfaces

We state necessary and sufficient conditions for a finite abelian group to act as a group of automorphisms of some compact bordered Klein surface of algebraic genus \(p>1\). This result provides a new method to obtain the real genus, as well as solve the maximum order problem, of abelian groups. We also compute the least real genus of abelian groups of the same order.

Automorphism groups of non-orientable elliptic-hyperelliptic Klein surfaces with boundary

A classical study about Klein and Riemann surfaces consists in determining their groups of automorphisms. This problem is very difficult in general,and it has been solved for particular families of surfaces or for fixed topological types. In this paper, we calculate the automorphism groups of non-orientable bordered elliptic-hyperelliptic Klein surfaces of algebraic genus $$p>5$$ .

Extensions of finite cyclic group actions on bordered surfaces

We study the question of the extendability of the action of a finite cyclic group on a compact bordered Klein surface (either orientable or non-orientable). This extends previous work by the authors for group actions on unbordered surfaces. It is shown that if such a cyclic action is realised by means of a non-maximal NEC signature, then the action always extends. For a given integer g ≥ 2 , we determine the order of the largest cyclic group that acts as the full automorphism group of a bordered surface of algebraic genus g, and the topological type of the surfaces on which the largest action takes place. In addition, we calculate the smallest algebraic genus of a bordered surface on which a given cyclic group acts as the full automorphism group of the surface. For this, we deal separately with orientable and non-orientable surfaces, and we also determine the topological type of the surfaces attaining the bounds.

The Density of the Real Genus Values of p-Groups

Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Let p be an odd prime, and let Jp be the set of integers g for which there is a p-group of real genus g. We show that the set Jp has density zero in the set of positive integers.

Groups of real genus p + 1, where p is prime

Let G be a finite group. The real genusρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We classify the large groups of real genus p + 1, that is, the groups such that |G| ≥ 3(g - 1), where the genus action of G is on a bordered surface of genus g = p + 1. The group G must belong to one of four infinite families. In addition, we determine the order of the largest automorphism group of a surface of genus g for all g such that g = p + 1, where p is a prime.

The Real Genus of <i>p</i>-Groups

Let G be a finite group. The real genus p(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We show that a non-cyclic p-group G has real genus not congruent to (1 — p3) (mod 2p3) if and only if G is in one of 14 families of groups. The genus formula for each of these 14 families of groups is determined. A consequence of this classification is that almost all positive integers that are the real genus of a p-group are congruent to (1 p3) (mod 2p3). Finally, the integers that occur as the real genus of a p-group with Frattini-class 2 have density zero in the positive integers.

Geometrical study of prime order automorphisms on Klein surfaces

We consider (n,q)-gonal Klein surfaces of algebraic genus p≥2 for which n≥p and n prime. We exclude the case n=2 corresponding to q-hyperelliptic surfaces, so n≥3. We study geometrical conditions on fundamental polygons of the NEC groups which uniformize these surfaces. From these conditions we give a way to construct such surfaces and obtain parameters which characterize them.

THE REAL GENUS OF 2-GROUPS II

Let G be a finite group. The real genus p(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We show that if G is a group of order 2m that has real genus congruent to 5 (mod 8), then either G has exponent 2m~3 and a dihedral subgroup of index 4 or else the exponent of G is 2m~2. It follows that there are at most 53 isomorphism types of 2-groups with real genus congruent to 5 (mod 8). A consequence of this bound is that almost all positive integers that are the real genus of a 2-group are congruent to 1 (mod 8). We also prove that almost all positive integers that are the real genus of a 2-group with Frattini-class 2 are in a set that has density zero in the positive integers.

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2 ≤ p ≤ 6 . We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p ≥ 2 .

On the connections between the topological genus of certain polyhedra and the algebraic genus of their Hilton-Hopf quadratic forms

The Hilton-Hopf quadratic form is defined for spaces of the homotopy type of a CW complex with one cell each in dimensions 0 and 4n, K cells in dimension 2n and no other cells. If two such spaces are of the same topological genus, then their Hilton-Hopf quadratic forms are of the same weak algebraic genus. For large classes of spaces, such as simply connected differentiable 4-manifolds, the converse is also true, as long as the suspensions of the spaces are also of the same topological genus. This note allays the conjecture that the converse is true in general by offering two techniques for generating infinite families of counterexamples.

Automorphism groups of q-trigonal planar Klein surfaces and maximal surfaces

A compact Klein surface X=D/Γ , where D denotes the hyperbolic plane and Γ is a surface NEC group, is said to be q-trigonal if it admits an automorphism ϕ of order 3 such that the quotient X/<ϕ > has algebraic genus q. In this paper we obtain for each q the automorphism groups of q-trigonal planar Klein surfaces, that is surfaces of topological genus 0 with k≥ 3 boundary components. We also study the surfaces in this family, which have an automorphism group of maximal order (maximal surfaces). It will be done from an algebraic and geometrical point of view.

The real genus of the alternating groups

A Klein surface with boundary of algebraic genus \mathfrak{p}\geq 2 , has at most 12(\mathfrak{p}-1) automorphisms. The groups attaining this upper bound are called M^{\ast} -groups, and the corresponding surfaces are said to have maximal symmetry. The M^{\ast} -groups are characterized by a partial presentation by generators and relators. The alternating groups A_{n} were proved to be M^{\ast} -groups when n\geq 168 by M. Conder. In this work we prove that A_{n} is an M^{\ast } -group if and only if n\geq 13 or n=5,10 . In addition, we describe topologically the surfaces with maximal symmetry having A_{n} as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for n\leq 14 . Each finite group G acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, \rho(G) . If G is an M^{\ast} -group then \rho(G)=\frac{o(G)}{12}+1 . We end our work by calculating the real genus of the alternating groups which are not M^{\ast} -groups.