Maximal group actions on compact oriented surfaces (original) (raw)

On Finite Groups of Symmetries of Surfaces

2013

The genus spectrum of a finite group GGG is the set of all ggeq2g\geq 2ggeq2 such that GGG acts faithfully and orientation-preserving on a closed compact orientable surface of genus ggg. This article is an overview of some results relating the genus spectrum of GGG to its group theoretical properties. In particular, the arithmetical properties of genus spectra are discussed, and explicit results are given on the 2-groups of maximal class, certain sporadic simple groups and a some of the groups PSL$(2,q)$, where qqq is a small prime power. These results are partially new, and obtained through both theoretical reasoning and application of computational techniques.

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

The Epstein Birthday Schrift

Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath's results to include the number of ovals and also determine whether they are twisted or not.

Cyclic p-groups of symmetries of surfaces

Glasgow Mathematical Journal, 1991

Let Σg denote a compact orientable surface of genus g ≥ 2. We consider finite groups G acting effectively on Σg and preserving the orientation—for short, G acts on Σg or Gis a symmetry group of Σg. Each surface Σg admits only finitely many symmetry groups G and the orders of these groups are bounded by Wiman's bound of 84(g – 1). This bound is attained for infinitely many values of g [12], see also [9], and all values of g ≤ 104 for which it is attained are known [4].

Periodic Automorphisms of Surfaces: Invariant Circles and Maximal Orders

Experimental Mathematics, 2000

CONTENTS 1. Introduction and Main Results 2. Periodic Automorphisms and Branched Covers 3. Automorphisms Without Invariant Circles 4. Nonorientable Surfaces 5. Maximal Orders of Periodic Automorphisms References W. H. Meeks has asked the following question: For what g does every (orientation preserving) periodic automorphism of a closed orientable surface of genus g have an invariant circle? A variant of this question due to R. D. Edwards asks for the existence of invariant essential circles. Using a construction of Meeks we show that the answer to his question is negative for all but 43 values of g 10000, all of which lie below g = 105. We then show that the work of S. C. Wang on Edwards' question generalizes to nonorientable surfaces and automorphisms of odd order. Motivated by this, we ask for the maximal odd order of a periodic automorphism of a given nonorientable surface. We obtain a fairly complete answer to this question and also observe an amusing relation between this order and Fermat primes.

AUTOMORPHISM GROUPS OF RIEMANN SURFACES OF GENUS p+1, WHERE p IS PRIME

Glasgow Mathematical Journal, 2005

We show that if S is a compact Riemann surface of genus g = p + 1, where p is prime, with a group of automorphisms G such that |G| ≥ λ(g − 1) for some real number λ > 6, then for all sufficiently large p (depending on λ), S and G lie in one of six infinite sequences of examples. In particular, if λ = 8 then this holds for all p ≥ 17.

Non-maximal cyclic group actions on compact Riemann surface

Revista Matemática Complutense, 1997

We say thai a finite group G of automarphisms of a Riemann surface X is norz-maximal in genus y if (i) G acta as a group of autamorphisnis of sorne compací Riemann surface X 9 of genus y ami <u), for alí sueh surfaces X9 , ¡ Ami X9 ¡>¡ 0. In ihis paper we investigate ihe case where O is a cyelie group O,. of arder n. If O,, acis on only finiiely niany surfaces of genus a, ihen we cornpletely salve ihe problem of flnding alí such pairs (ri,g).

Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries

Progress in Mathematics, 2000

Let S g,b,p denote a surface which is connected, orientable, has genus g, has b boundary components, and has p punctures. Let Σ g,b,p denote the fundamental group of S g,b,p . Let Out g,b,p denote the algebraic mapping-class group of S g,b,p .

On finite abelian ppp-groups of symmetries of Riemann surfaces

arXiv: Group Theory, 2016

The genus spectrum of a finite group GGG is the set of all ggg such that GGG acts faithfully on a compact Riemann surface of genus ggg. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the abelian ppp-groups. Motivated by the work of Talu for odd primes ppp, we develop a general combinatorial machinery, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of abelian ppp-groups. We have a particular view towards how to generally find the reduced minimum genus in this class of groups, determine the complete genus spectrum for a large subclass of abelian ppp-groups, consisting of those groups in a certain sense having `large' defining invariants, and use this to construct infinitely many counterexamples to Talu's Conjecture, saying that an abelian ppp-group is recoverable from its genus spectrum. Finally, we indicate the effectiveness of our combinatorial approach ...