On the fixed-point set of automorphisms of non-orientable surfaces without boundary

Izquierdo, Milagros; Singerman, David

doi:10.2140/gtm.1998.1.295

Cited by 8 publications

(5 citation statements)

References 10 publications

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“…Harvey's theorem provides necessary and sufficient conditions for the abstract Fuchsian group to be a universal covering transformation group of the cyclic group, while Macbeath gives a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of compact Riemann surface. The generalization of that formula to closed non-orientable surfaces was obtained by Izquierdo and Singerman [6].…”

Section: Introductionmentioning

confidence: 93%

Sets of periods for automorphisms of compact Riemann surfaces

Sierakowski¹

2007

Journal of Pure and Applied Algebra

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Let G = f be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g ≥ 2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x ∈ S, x is fixed by f d but not by any smaller power of f . For an arbitrary subset A of proper divisors of N , there is always an associated action and, if g A denotes the minimal genus for such an action, an algorithm is obtained here to determine g A . Furthermore, a set A max is determined for which g A is maximal.

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Section: Introductionmentioning

confidence: 93%

Sets of periods for automorphisms of compact Riemann surfaces

Sierakowski¹

2007

Journal of Pure and Applied Algebra

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“…We mention, however, that some attempts in this direction have been made by Sierakowski in his thesis [16]. Finally, we emphasize that to find the whole conformal dynamics on Klein surfaces seems to be a difficult problem, though its special form, consisting in finding, in terms of the group of automorphisms of the surface and the topological type of the action, the set of fixed points of a single automorphism has been solved in [4,6] for bordered and non-orientable unbordered Klein surfaces, respectively, and in [5,9,13] for classical Riemann surfaces in the case of symmetries and non-involutionary automorphisms respectively.…”

Section: Introductionmentioning

confidence: 99%

Conformal actions with prescribed periods on Riemann surfaces

Gromadzki

Marzantowicz

2011

Fund. Math.

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It is a natural question what is the set of minimal periods of a holomorphic maps on a Riemann surface of negative Euler characteristic. Sierakowski studied ordinary holomorphic periods on classical Riemann surfaces. Here we study orientation reversing automorphisms acting on classical Riemann surfaces, and also automorphisms of nonorientable unbordered Klein surfaces to which, following Singerman, we shall refer to as non-orientable Riemann surfaces. We get a complete set of conditions for the existence of conformal actions with a prescribed order and a prescribed set of periods together with multiplicities. This lets us determine the minimal genus of a surface which admits such an action.

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“…Macbeath found in [11] a formula for the number of points fixed by an arbitrary analytic automorphism ϕ of a compact Riemann surface X in terms of the topological type of the group G of conformal automorphisms of X, which corresponds to a so called smooth epimorphism θ : Λ → G. Later, Izquierdo and Singerman [9] obtained in such terms formulae for G a cyclic group of automorphisms of an unbordered non-orientable compact Klein surface, and Gromadzki [8] provided such formulae for arbitrary group G of all dianalytic automorphisms of X. Here we deal with the case in which X is a compact bordered Klein surface.…”

Section: Introductionmentioning

confidence: 99%

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

Gamboa¹,

Gromadzki²

2012

Rev. Mat. Iberoam.

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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.

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On the fixed-point set of automorphisms of non-orientable surfaces without boundary

Cited by 8 publications

References 10 publications

Sets of periods for automorphisms of compact Riemann surfaces

Sets of periods for automorphisms of compact Riemann surfaces

Conformal actions with prescribed periods on Riemann surfaces

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

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