Klein Surfaces Research Articles

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.

On the connectedness of the branch loci of moduli spaces of orientable Klein surfaces

Let \(\mathcal {M}_{(g,+,k)}^{K}\) be the moduli space of orientable Klein surfaces of genus \(g\) with \(k\) boundary components (see Alling and Greenleaf in Lecture notes in mathematics, vol 219. Springer, Berlin, 1971; Natanzon in Russ Math Surv 45(6):53–108, 1990). The space \(\mathcal {M}_{(g,+,k)} ^{K}\) has a natural orbifold structure with singular locus \(\mathcal {B} _{(g,+,k)}^{K}\). If \(g>2\) or \(k>0\) and \(2g+k>3\) the set \(\mathcal {B} _{(g,+,k)}^{K}\) consists of the Klein surfaces admitting non-trivial symmetries and we prove that, in this case, the singular locus is connected.

Associating divisors with quadratic differentials on Klein surfaces

We extend the notion and basic properties of quadratic differential divisors to a Klein surface, using the corresponding form’s divisors from the double cover of the Klein surface.

Non-rationality of some fibrations associated to Klein surfaces

We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic fibre of the polynomial fibration is a surface defined over the function field of the line. We proved that it is not rational in cases D, E, although it is obviously rational in the case A. The group of automorphisms of the Klein surfaces is also described, and is linear and of finite dimension in cases D, E; this result being obviously false in case A.

Abelian Yang-Mills Theory on Real Tori and Theta Divisors of Klein Surfaces

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.

The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 3, 4 and 5

An important problem in the study of Riemann and Klein surfaces is determining their full automorphism groups. Up to now only very partial results are known, concerning surfaces of low genus or families of surfaces with special properties. This paper deals with non-orientable unbordered Klein surfaces. In this case the solution of the problem is known for surfaces of genus 1 and 2, and for hyperelliptic surfaces. Here we explicitly obtain the full automorphism group of all surfaces of genus 3, 4 and 5.

On automorphisms of Klein surfaces with invariant subsets

On automorphisms of Klein surfaces with invariant subsets

Compact Klein surfaces of genus 5 with a unique extremal disc

A compact (orientable or non-orientable) surface of genus g g is said to be extremal if it contains an extremal disc, that is, a disc of the largest radius determined only by g g . The present paper concerns non-orientable extremal surfaces of genus 5 5 . We represent the surfaces as side-pairing patterns of a hyperbolic regular 24 24 -gon, that is, a generic fundamental region of an NEC group uniformizing each of the surfaces. We also describe the group of automorphisms of the surfaces with a unique extremal disc.

Lowest uniformizations of compact Klein surfaces

A Schottky group is a purely loxodromic Kleinian group, with a nonempty region of discontinuity, isomorphic to a free group of finite rank. An extended Schottky group is an extended Kleinian group whose orientation-preserving half is a Schottky group. The collection of uniformizations of either a closed Riemann surface or a compact Klein surface is partially ordered. In the case of closed Riemann surfaces, the lowest uniformizations are provided by Schottky groups. In this paper we provide simple arguments to see that the lowest uniformizations of compact Klein surfaces are exactly those produced by extended Schottky groups.

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.

THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

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 crosscap number and denoted by [Formula: see text]. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with [Formula: see text] is finite. The value of [Formula: see text] is known when G belongs to certain families of groups. In particular, if o(G) < 32, [Formula: see text] is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.

The real genus of the groups of order 48

Every finite group acts as an automorphism group of several bordered compact Klein surfaces. The minimal genus of these surfaces is called the real genus of and it is denoted The systematical study of this parameter was begun by May and continued by him in several papers about the topic. As a consequence of these works, he and other authors obtained the groups such that On the other hand, the real genus of the groups of order less than 64, excepting 48, is already known. The orders 48 and 64 are just the first obstructions in the search of the groups of real genus greater than 16. In this work we complete the calculation of the real genus of all the fifty two groups of order 48.

Moduli spaces of Klein surfaces and related operads

We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul and we identify the dual dg operad governing A-infinity algebras with involution in terms of Mobius graphs which are a generalisation of ribbon graphs. We then generalise open topological conformal field theories to open Klein topological conformal field theories and give a generators and relations description of the open KTCFT operad. We deduce an analogue of the ribbon graph decomposition of the moduli spaces of Riemann surfaces: a Mobius graph decomposition of the moduli spaces of Klein surfaces (real algebraic curves). The Mobius graph complex then computes the homology of these moduli spaces. We also obtain a different graph complex computing the homology of the moduli spaces of admissible stable symmetric Riemann surfaces which are partial compactifications of the moduli spaces of Klein surfaces.

PRIME ORDER AUTOMORPHISMS OF KLEIN SURFACES REPRESENTABLE BY ROTATIONS ON THE EUCLIDEAN SPACE

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S, f) to be conformally equivalent to (S′, f′) where S′ is a bordered orientable Klein surface embedded in the Euclidean space and f′ is the restriction to S′ of a prime order rotation. Our results can allow the visualization of some Riemann surfaces automorphisms by representing them as restrictions of isometries of 𝕊4 or ℝ4. We illustrate this method with the order seven automorphisms of two famous Riemann surfaces: the Klein quartic and the Wiman surface.

DOUBLES OF KLEIN SURFACES

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

Symmetric theta divisors of Klein surfaces

This is a slightly expanded version of the talk given by the first author at the conference Instantons in complex geometry, at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces. In this paper we compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.

On symmetric representations of groups of automorphism of bordered Klein surfaces

An automorphism of a bordered compact Klein surface induces a permutation of its boundary components and we study the corresponding representations of groups of automorphisms of such surfaces in the corresponding finite symmetric groups. The principal result gives a characterization of those abstract representations of a finite group G in symmetric groups which, up to the natural equivalence, proceed from compact Klein surfaces on which G acts as a group of dianalytic automorphisms. We deal with such representations for actions of G given by a so called smooth epimorphisms Λ → G, where Λ are so called non-Euclidean crystallographic groups (NEC-groups). For such data we calculate the kernels of our representations which allow, as an application, to get some results on the minimal degree of such representations for finite abstract groups.

On the set of fixed points of automorphisms of bordered Klein surfaces

The nature of the set of points fixed by automorphisms of Riemann or unbordered nonorientable Klein surfaces as well as quantitative formulae for them were found by Macbeath, Izquierdo, Singerman and Gromadzki in a series of papers. The possible set of points fixed by involutions of bordered Klein surfaces has been found by Bujalance, Costa, Natanzon and Singerman who showed that it consists of isolated fixed points, ovals and chains of arcs. They classified involutions of such surfaces, up to topological conjugacy in these terms. Here we give formulae for the number of elements of each type, also for non-involutory automorphisms, in terms of the topological type of the action of the group of dianalytic automorphisms. Finally we give some illustrative examples concerning bordered Klein surfaces with large groups of automorphisms already considered by May and Bujalance.

Real Determinant Line Bundles

This article is an expanded version of the talk given by Ch. O. at the Second Latin Congress on Symmetries in Geometry and Physics in Curitiba, Brazil in December 2010. In this version we explain the topological and gauge-theoretical aspects of our paper Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces, Comm. Math. Phys. Vol. 323, 813-858 (2013)

The framed little 2-discs operad and diffeomorphisms of handlebodies

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (that is, the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ‘open string’ modular operad made from moduli spaces of Riemann surfaces with marked intervals on the boundary. Our technique also recovers a theorem of Braun that the derived modular envelope of the cyclic operad that describes associative algebras with involution is homotopy equivalent to the modular operad made from moduli spaces of unoriented Klein surfaces with open string gluing.