Parametrization of the Teichmüller space of bordered surface NEC groups

A non-Euclidean crystallographic group F （NEC group, for short） is a discrete subgroup of isometries of the hyperbolic plane H, with compact quotient space H/Г. These groups uniformize Klein surfaces, surfaces endowed with dianalytic structure. These surfaces can be seen as a generalization of Riemann surfaces.
Fundamental polygons play an important role in the study of parametrizations of the Teichmuller space of NEC groups.
In this work we construct a class of right-angled polygons which are fundamental regions of bordered surface NEC groups. The free parameters used in the construction of the polygons give a parametrization of the Teichmuller space. From the parameters we obtain explicit matrices of the generators of the groups. Finally, we give examples to exhibit how different relations between the parameters reflect the existence of automorphisms on the quotient surfaces.     

On abelian automorphism group of a surface of general type

Summary LetS be a minimal surface of general type over,K the canonical divisor ofS. LetG be an abelian automorphism group ofS. IfK ²140, then the order ofG is at most 52K ²+32. Examples are also provided with an abelian automorphism group of order 12K ²+96.The automorphism groups for a complex algebraic curve of genusg2 have been thoroughly studied by many authors, including many recent ones. In particular, various bounds have been established for the order of such groups: for example, the order of the total automorphism group is 84(g–1) [Hu], that of an abelian subgroup is 4g+4 [N], while the order of any automorphism is 4g+2 ([W], see also [Ha]).It is an intriguing problem to generalise these bounds to higher dimensions. For example, for surfaces of general type, it is well known that the automorphism groups are finite, and the bound of the orders of these groups depends only on the Chern numbers of the surface [A].In the attempts to such generalisations, the order of abelian subgroups has a special importance. Due to Jordan's theorem on group representations (and its followers), a bound on the order of abelian subgroups induces a bound on that of the whole automorphism group, although bounds thus obtained are generally far from satisfactory. In [H-S], it is shown that for surfaces of general type, the order of such an abelian subgroup is bounded by the square of the Chern numbers times a constant.The purpose of this article is to give a further analysis to the abelian case for surfaces of general type, in proving that the order is bounded linearly by the Chern numbers of the surface, in good analogy with the case of curves. More precisely, our main result is the following.Oblatum 11-IX-1989 & 29-I-1990     

Surfaces de la classe VII0 et automorphismes

In this Note we construct surfaces S of class VII containing a global spherical shell starting with a Hénon automorphism H(x,y)=(x²+c−ay,x) of C². For a special choice of the parameter a we show the existence of a non-trivial holomorphic vector field X on S. This contributes to the (open) problem of classifying all compact complex surfaces with a positive dimensional automorphism group. For general results concerning vector fields and foliations on surfaces containing a global spherical shell we refer to our note [2].     

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).     

On the topological type of anticonformal square roots of automorphisms of Riemann surfaces

Let X be a compact Riemann surface and f be a conformal automorphism of X of order n. An anticonformal square root of f is an anticonformal automorphism g of X such that g²=f. If g₁ and g₂ are two anticonformal square roots of f we study the problem of whether g₁ and g₂ have the same topological type, i. e., if there exists a homeomorphism h:X→X such that g₁=hg₂h⁻¹. We use techniques of noneuclidean crystallographic (NEC) groups and the topological classification of orientation reversing maps of finite period on surfaces given in [C1] and [Y]. Partially supported by DGICYT PB92-0716 and EC project CHRX-CT93-408     

REAL BELYI THEORY

We develop a Belyi-type theory that applies to Klein surfaces,that is, (possibly non-orientable) surfaces with boundary whichcarry a dianalytic structure. In particular, we extend Belyi'sfamous theorem from Riemann surfaces to Klein surfaces.     

Applications of a theorem of Singerman about Fuchsian groups

Assume that we have a (compact) Riemann surface S, of genus greater than 2, with , where is the complex unit disc and Γ is a surface Fuchsian group. Let us further consider that S has an automorphism group G in such a way that the orbifold S/G is isomorphic to where is a Fuchsian group such that and has signature σ appearing in the list of non-finitely maximal signatures of Fuchsian groups of Theorems 1 and 2 in [6]. We establish an algebraic condition for G such that if G satisfies such a condition then the group of automorphisms of S is strictly greater than G, i.e., the surface S is more symmetric that we are supposing. In these cases, we establish analytic information on S from topological and algebraic conditions. Received: 4 April 2008     

Automorphism groups of the real projective plane with holes and their conjugacy classes within its mapping class group

For each integer g2 we give the complete list of groups acting as a group of dianalytic automorphisms of a real projective plane with g holes, which, in algebraic terms, correspond to birational automorphisms of real algebraic (M–1)-curves. We also determine those acting as the full group of automorphisms of such a surface. Furthermore, the conjugacy classes of the finite subgroups of its mapping class group are calculated.Mathematics 2000 Subject Classification (2000): 30F, 32G, 14H.Partially supported by BFM2002-04801.Partially supported by BFM2002-04801 and RAAG HPRN-CT-2001-00271.Partially supported by GAAR BFM2002-04797 and RAAG HPRN-CT-2001-00271     

Symmetries of Accola-Maclachlan and Kulkarni surfaces

For all there is a Riemann surface of genus whose automorphism group has order , establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large the Accola-Maclachlan surface was unique for and produced exactly one additional surface (the Kulkarni surface) for . In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.

     

On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.     

A non-Abelian K-theory and pseudo-isotopies of 3-manifolds

A non-Abelian version of algebraic K-theory, based on automorphism of free products rather than automorphisms of free modules, is considered and is related to pseudo-isotopies of 3-manifolds.Sometime after writing this paper I learned that some of the algebraic results in it were first proved by Gersten in [13]. Specifically each of Theorems 3.1, 3.2 and 5.1 in the special case when is the trivial group, and Theorem 3.3 and its corollaries are all results of Gersten. I have left the paper as it is for the sake of completeness and since the approach here often differs considerably from Gersten's.     

Automorphism groups of compact bordered Klein surfaces with invariant subsets

In this work we get upper bounds for the order of a group of automorphisms of a compact bordered Klein surface S of algebraic genus greater than 1. These bounds depend on the algebraic genus of S and on the cardinals of finite subsets of S which are invariant under the action of the group. We use our results to obtain upper bounds for the order of a group of automorphism whose action on the set of connected components of the boundary of S is not transitive. The bounds obtained this way depend only on the algebraic genus of S. The author is partially supported by the European Network RAAG HPRN-CT-2001-00271 and the Spanish GAAR DGICYT BFM2002-04797.     

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 35)

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group PL(2, 3⁵).     

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.     

Four-dimensional compact projective planes with a nonsolvable automorphism group

We contribute to the enumeration of all four-dimensional compact projective planes with an at least seven-dimensional automorphism group (cf. Betten [8]) by treating the nonsolvable case. Moreover, we find that the only possible six-dimensional nonsolvable automorphism group is ² · GL ₊ ² .Dedicated to Professor H. Salzmann on his 60th birthday     

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Let X be an algebraic surface defined over the complex field and endowed with an -fibration. As such a surface we have a Platonic -fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed -action and its fixpoint deleted off. We consider an étale endomorphism and show that is an automorphism in most cases. Of particular interest is the case of a Platonic -fiber space, for which being an automorphism is closely related to the Jacobian Problem for the affine plane . We also investigate the automorphism group of such surfaces. Received January 7, 1999 / Published online October 30, 2000     

On the essential dimension of a finite group

Let f(x) = aixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilberts 13th problem.Our approach to this question (and generalizations) is basedon the idea of the essential dimension of a finite group G:the smallest possible dimension of an algebraic G-variety over k to which one can compress a faithful linear representation of G. We show that dk(n) is just the essential dimension of the symmetricgroup Sn. We give results on the essential dimension ofother groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.     

Crossed products and entropy of automorphisms

Let be an exact C∗-algebra, let G be a locally compact group, and let be a C∗-dynamical system. Each automorphism α_g induces a spatial automorphism Ad_λg on the reduced crossed product . In this paper we examine the question, first raised by E. Størmer, of when the topological entropies of α_g and Ad_λg coincide. This had been answered by N. Brown for the particular case of discrete abelian groups. Using different methods, we extend his result to preservation of entropy for α_g when the subgroup of Aut(G) generated by the corresponding inner automorphism Ad_g has compact closure. This property is satisfied by all elements of a wide class of groups called locally [FIA]⁻. This class includes all abelian groups, both discrete and continuous, as well as all compact groups.     

Continuous families of rational surface automorphisms with positive entropy

For any k we construct k-parameter families of rational surface automorphisms with positive entropy. These are automorphisms of surfaces ${\mathcal{X}}$ , which are constructed from iterated blowups over the projective plane. In certain cases we are able to determine the exact automorphism group of ${\mathcal{X}}$ , as well as when two of these surfaces are inequivalent.