We classify compact Riemann surfaces of genus g, where g−1 is a prime p, which have a group of automorphisms of order ρ(g−1)for some integer ρ≥1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ>6, and of the first and third authors for ρ= 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p+1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

In this article we determine the maximal possible order of the automorphism group 18 of the form ag + b, where a and b are integers, of a complex three and four- 19 dimensional family of compact Riemann surfaces of genus g, appearing for all genus.In addition, we construct and describe explicit complex three and four-dimensional 20 families possessing these maximal numbers of automorphisms.

Belolipetsky and Jones classified those compact Riemann surfaces of genus g admitting a large group of automorphisms of order lambda(g - 1), for each lambda amp;gt; 6, under the assumption that g - 1 is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting 5(g - 1) and 6(g - 1) automorphisms, with g - 1 a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order 3(g - 1), with g - 1 a prime number. We also provide isogeny decompositions of their Jacobian varieties. (C) 2019 Elsevier Inc. All rights reserved.

We classify compact Riemann surfaces of genus g, where _g−1 is a prime p, whichhave a group of automorphisms of order ρ(g−1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3,4,5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p +1, together with the non-orientable regular hypermaps of characteristic − p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

Given a compact Riemann surface X with an action of a finite group G, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We consider the set of equisymmetric Riemann surfaces M(2n -1, D-2n, theta) for all n amp;gt;= 2. We study the group algebra decomposition of the Jacobian JX of every curve X is an element of M (2n - 1, D-2n, theta) for all admissible actions, and we provide affine models for them. We use the topological equivalence of actions on the curves to obtain facts regarding its Jacobians. We describe some of the factors of JX as Jacobian (or Prym) varieties of intermediate coverings. Finally, we compute the dimension of the corresponding Shimura domains.

Unfortunatelly an erratum appears in the foreword of this volume.

We claim that one of the authors of the article On universal groups and three-manifolds, Invent. Math. 87 (1987), no. 3, 441–456 is W. Witten. The correct name of this author is Wilbur Carrington Whitten. W. C. Whitten is not the father of the fields medallist E. Witten, as we claim there.

Schottky space Sg is the space that parametrizes PSL2(C)-conjugacy classes of Schottky groups of rank g ≥ 2. The branch locus Bg consists of the conjugacy classes of those Schottky groups which are a finite index proper subgroup of some Kleinian group. In a previous paper we observed that Bg was connected for g ≥ 3 odd and that it has at most two components for g ≥ 4 even. In this short note, we observe that Bg is always connected.

We prove that themaximal number ag+b of automorphisms of equisymmetric and

complex-uniparametric families of Riemann surfaces appearing in all genera is 4g + 4. For

each integer g ≥ 2 we find an equisymmetric complex-uniparametric family Ag of Riemann

surfaces of genus g having automorphism group of order 4g + 4. For g ≡ −1mod 4 we

present another uniparametric family Kg with automorphism group of order 4g + 4. The

family Ag contains the Accola–Maclachlan surface and the family Kg contains the Kulkarni

surface

María Teresa, Maite, Lozano is a great person and mathematician, in these pages we can only give a very small account of her results trying to resemble her personality. We will focus our attention only on a few of the facets of her work, mainly in collaboration with Mike Hilden and José María Montesinos because as Maite Lozano pointed out in an international conference in Umeå University in June 2017, where she was a plenary speaker:

I am specially proud of been part of the team Hilden-Lozano-Montesinos (H-L-M), and of our mathematical achievements

We determine, for all genus g≥2g≥2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g  ≠ 3,6,12,153,6,12,15 or 30, these surfaces form a real Riemann surface F_gFg in the moduli space M_gMg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real forms of real Riemann surfaces in F_gFg. The set of real Riemann surfaces in F_gFg consists of three intervals its closure in the Deligne–Mumford compactification of M_gMg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in F_g

We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configuration in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on three distinct dianalytic surfaces of genus 57. We also give a geometric realisation of the latter pentagonal geometry in terms of points and hyperspheres in 24 dimensional Euclidean space. From these, we also obtain geometric realisations in terms of points and circles (or hyperspheres) of pentagonal geometries with k circles (hyperspheres) through each point and k 1 points on each circle (hypersphere).

We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with a maximum number of automorphisms for their genus, and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle v4'>v4v4 configurations on p-gonal surfaces with two p-gonal morphisms is given. The image of these configurations on the sphere under the two p-gonal morphisms is also described.

Consider the moduli space M g of Riemann surfaces of genusg≥2 and its Deligne-Munford compactification M g ¯ . We are interested in the branch locus B g for g>2 , i.e., the subset of M g consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in M g but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for g≥5 the set of (cyclic) trigonal surfaces is connected in M g ¯ . To do so we exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of 3-gonal Riemann surfaces. For p>3 the connectivity of the p -gonal loci becomes more involved. We show that for p≥11 prime and genus g=p−1 there are one-dimensional strata of cyclic p -gonal surfaces that are completely isolated in the completion B g ¯ of the branch locus in M g ¯ .

In this paper we study the automorphism groups of real curves admitting a regular meromorphic function f of degree p, so called real cyclic p-gonal curves. When p = 2 the automorphism groups of real hyperelliptic curves where given by Bujalance et al. in [4].

Let M K (g,+,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 M K (g,+,k) has a natural orbifold structure with singular locus B K (g,+,k) . If g>2 or k>0 and 2g+k>3 the set B K (g,+,k) consists of the Klein surfaces admitting non-trivial symmetries and we prove that, in this case, the singular locus is connected.

Let M be a handlebody of genus g greater than= 2. The space T(M), that parametrizes marked Kleinian structures on M up to isomorphisms, can be identified with the space MSg, of marked Schottky groups of rank g, so it carries a structure of complex manifold of finite dimension 3(g - 1). The space M(M) parametrizing Kleinian structures on M up to isomorphisms, can be identified with S-g, the Schottky space of rank g, and it carries the structure of a complex orbifold. In these identifications, the projection map pi: T(M) -greater than M(M) corresponds to the map from MSg, onto S-g that forgets the marking. In this paper we observe that the singular locus B(M) of M(M), that is, the branch locus of pi, has (i) exactly two connected components for g = 2, (ii) at most two connected components for g greater than= 4 even, and (iii) M(M) is connected for g greater than= 3 odd.

The moduli space M_g, of compact Riemann surfaces of genus g has orbifold structure since M_g is the quotient space of the Tiechmüller space by the action of the mapping class group. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space we find the orbifold structure of the moduli spaces of Riemann surfaces of genera 4 and 5.

In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p⩾3p⩾3 is a prime integer and the genus of the surfaces is at least (p−1)²+1(p−1)2+1. We use Fuchsian and NEC groups, and cohomology of finite groups.

The moduli space M-g of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of the orbifold is the branch locus B-g. In this article we show that B-g is connected for genera three, four, thirteen, seventeen, nineteen and fiftynine, and disconnected for any other genus. In order to prove this we use Fuchsian groups, automorphisms of order 5 and 7 of Riemann surfaces, and calculations with GAP for some small genera.

Let be an integer and let , where denotes the moduli space of compact Riemann surfaces of genus . Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belong to the same connected component. We also prove the connectedness of for and with the exception of the isolated points given by Kulkarni.

 Let g be an integer >= 3 and let B-g = {X is an element of mu(g) : Aut(X) not equal Id} be the branch locus of mu(g), where mu(g) denotes the moduli space of compact Riemann surfaces of genus g. The structure of B-g is of substantial interest because B-g corresponds to the singularities of the action of the modular group on the Teichmuller space of surfaces of genus g (see [14]).

Kulkarni ([15], see also [13]) proved the existence of isolated points in the branch loci of the moduli spaces of Riemann surfaces. In this work we study the isolated connected components of dimension 1 in such loci. These isolated components of dimension one appear if the genus is g = p - 1 with p prime >= 11. We use uniformization by Fuchsian groups and the equisymmetric stratification of the branch loci.

n/a

Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, fractal geometry, conformal dynamics, discrete groups, geometric group theory, algebraic curves and their moduli, various kinds of deformation theory, coding, thermodynamic formalism, and topology of three-dimensional manifolds. This collection of articles, authored by leading authorities in the field, comprises 16 expository essays presenting original research and expert surveys of important topics related to Riemann surfaces and their geometry. It complements the body of recorded research presented in the primary literature by broadening, re-working and extending it in a more focused and less formal framework, and provides a valuable commentary on contemporary work in the subject. An introductory section sets the scene and provides sufficient background to allow graduate students and research workers from other related areas access to the field.

In this paper we find the maximal order of an automorphism of a trigonal Riemann surface of genus g, g5. We find that this order is smaller for generic than for cyclic trigonal Riemann surfaces, showing that generic trigonal surfaces have “less symmetry” than cyclic trigonal surfaces. Finally we prove that the maximal order is attained for infinitely many genera in both the cyclic and the generic case.

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.

The moduli space M-g of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus B-g. In this article we present some results related with the topology of B-g. We study the connectedness of B-g for g andlt;= 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces considered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

Let g be an integer ≥ 3 and let θg = {X ∈ M_g|Aut(X) ≠ 1_d}, where Mg denotes the moduli space of a compact Riemann surface. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belongs to the same connected component. We also prove the connectedness of θg for g = 5, 6, 7 and 8 with the exception of the isolated points given by Kulkarni.

A closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann Surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

Hurwitz spaces are spaces of pairs (S, f) where S is a Riemann surface and f : S ? C^ a meromorphicfunction. In this work, we study 1-dimensional Hurwitz spaces HDp of meromorphic p-fold functions with four branched points, three of them fixed, the corresponding monodromy representation over each branched point is a product of (p - 1)/2 transpositions and the monodromy groupis the dihedral group Dp. We prove that the completion HDp of the Hurwitz space HDp is uniformized by a non-nomal index p + 1 subgroup of a triangular group with signature (0, [p, p, p]). We also establish the relation of the meromorphic covers with elliptic functions and show that HDp is aquotient of the upper half plane by the modular group G (2) n G0 (p). Finally, we study the real forms of the Belyi projection HDp ? C^ and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.

A closed Riemann surface which is a 3-sheeted regular covering of the Riemann sphere is called cyclic trigonal, and such a covering is called a cyclic trigonal morphism. Accola showed that if the genus is greater or equal than 5 the trigonal morphism is unique. Costa-Izquierdo-Ying found a family of cyclic trigonal Riemann surfaces of genus 4 with two trigonal morphisms. In this work we show that this family is the Riemann sphere without three points. We also prove that the Hurwitz space of pairs (trigonal morphism, is the Riemann sphere with four punctures. Finally, we give the equations of the curves in the family.

    'Groups St Andrews 2005' was held in the University of St Andrews in August 2005 and this first volume of a two-volume book contains selected papers from the international conference. Four main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the Proceedings. This volume contains the contributions by Peter Cameron (Queen Mary, London) and Rostislav Grogorchuk (Texas A&M, USA).  Apart from the main speakers, refereed survey and research articles were contributed by other conference participants. Arranged in alphabetical order, these articles cover a wide spectrum of modern group theory. The regular Proceedings of Groups St Andrews conferences have provided snapshots of the state of research in group theory throughout the past 25 years. Earlier volumes have had a major impact on the development of group theory and it is anticipated that this volume will be equally important.

Let Gamma be a non-Euclidean crystallographic group. Gamma is said to be non-maximal if there exists a non-Euclidean crystallographic group Gamma' such that Gamma <= Gamma' and the dimension of the Teichmuller space of Gamma equals the dimension of the Teichmuller space of V. The full list Of such pairs of groups is computed in the case when Gamma is non-normal in Gamma'. The corresponding problem for Fuchsian groups was solved by Singerman.

A closed Riemann surface X which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface X is called real trigonal if there is an anticonformal involution (symmetry) a of X commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry or is the number of connected components of the fixed point set Fix(sigma) and the orientability of the Klein surface X/(sigma). We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. A p-gonal Riemann surface is called real p-gonal if there is an ant iconformal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regular covering the Riemann surface is called real cyclic p-gonal, otherwise it is called real generic p-gonal. The species of the symmetry σ is the number of connected components of the fixed point set Fix (σ) and the orientability of the Klein surface X/〈σ〉. In this paper we find the species for the possible symmetries of real cyclic p-gonal Riemann surfaces by means of Fuchsian and NEC groups.