scholarly journals Faithfulness of Actions on Riemann-Roch Spaces

2015 ◽  
Vol 67 (4) ◽  
pp. 848-869 ◽  
Author(s):  
Bernhard Köck ◽  
Joseph Tait
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Coding Theory ◽  
Algebraic Curve ◽  
Deformation Theory ◽  
Analogous Problem ◽  
Invariant Divisor ◽  
A Finite Group

AbstractGiven a faithful action of a finite groupGon an algebraic curveXof genusgX≥ 2, we giveexplicit criteria for the induced action ofGon the Riemann–Roch spaceH0(X,OX(D)) to be faithful,whereDis aG-invariant divisor on X of degree at least 2gX− 2. This leads to a concise answer to the question of when the action ofGon the spaceH0(X,Ωx⊗m) of global holomorphic polydifferentials of order m is faithful. IfXis hyperelliptic, we provide an explicit basis of H0(X,Ωx⊗m). Finally, we giveapplications in deformation theory and in coding theory and discuss the analogous problem for theaction ofGon the first homologyH1(X,ℤ/mℤ) ifXis a Riemann surface.

scholarly journals Equivariant splitting of the Hodge–de Rham exact sequence

2021 ◽  
Author(s):  
Jędrzej Garnek
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Short Exact Sequence ◽  
Algebraic Curve ◽  
Hyperelliptic Curves ◽  
Witt Vectors ◽  
De Rham ◽  
A Finite Group

AbstractLet X be an algebraic curve with a faithful action of a finite group G over a field k. We show that if the Hodge–de Rham short exact sequence of X splits G-equivariantly then the action of G on X is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.

scholarly journals A STRUCTURED DESCRIPTION OF THE GENUS SPECTRUM OF ABELIAN p-GROUPS

2018 ◽  
Vol 61 (2) ◽  
pp. 381-423
Author(s):  
JÜRGEN MÜLLER ◽  
SIDDHARTHA SARKAR
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Open Problem ◽  
Compact Riemann Surface ◽  
General Description ◽  
Combinatorial Method ◽  
A Finite Group ◽  
Almost All

AbstractThe genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.

scholarly journals THE SYMMETRIC GENUS OF 2-GROUPS

2012 ◽  
Vol 55 (1) ◽  
pp. 9-21 ◽  
Author(s):  
COY L. MAY ◽  
JAY ZIMMERMAN
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
A Finite Group ◽  
Almost All ◽  
Positive Integers

AbstractLet G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2m−3 and a dihedral subgroup of index 4 or else the exponent of G is 2m−2. We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

THE 2-GROUPS OF ODD STRONG SYMMETRIC GENUS

2010 ◽  
Vol 09 (03) ◽  
pp. 465-481 ◽  
Author(s):  
COY L. MAY ◽  
JAY ZIMMERMAN
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
A Finite Group ◽  
Almost All ◽  
Positive Integers

Let G be a finite group. The strong symmetric genusσ0(G) is the minimum genus of any Riemann surface on which G acts preserving orientation. We show that a 2-group G has strong symmetric genus congruent to 3 (mod 4) if and only if G is in one of 14 families of groups. A consequence of this classification is that almost all positive integers that are the genus of a 2-group are congruent to 1 (mod 4).

On the Monodromy Groups of Riemann Surfaces of Genus ≧1

1981 ◽  
Vol 33 (5) ◽  
pp. 1142-1156
Author(s):  
Kathryn Kuiken
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Abelian Groups ◽  
General Question ◽  
Group Of Automorphisms ◽  
Cyclic Groups ◽  
Monodromy Groups ◽  
Groups Of Automorphisms ◽  
A Finite Group

It is well-known [5, 19] that every finite group can appear as a group of automorphisms of an algebraic Riemann surface. Hurwitz [9, 10] showed that the order of such a group can never exceed 84 (g – 1) provided that the genus g is ≧2. In fact, he showed that this bound is the best possible since groups of automorphisms of order 84 (g – 1) are obtainable for some surfaces of genus g. The problems considered by Hurwitz and others can be considered as particular cases of a more general question: Given a finite group G, what is the minimum genus of the surface for which it is a group of automorphisms? This question has been completely answered for cyclic groups by Harvey [7]. Wiman's bound 2(2g + 1), the best possible, materializes as a consequence. A further step was taken by Maclachlan who answered this question for non-cyclic Abelian groups.

scholarly journals The symmetric genus of 2-groups

1999 ◽  
Vol 41 (1) ◽  
pp. 115-124 ◽  
Author(s):  
JAY ZIMMERMAN
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Group Of Automorphisms ◽  
A Finite Group

A finite group G can be represented as a group of automorphisms of a compact Riemann surface, that is, G acts on a Riemann surface. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts (possibly reversing orientation).

Extending finite group actions on surfaces to hyperbolic 3-manifolds

1995 ◽  
Vol 117 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Monique Gradolato ◽  
Bruno Zimmermann
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Finite Groups ◽  
Group Actions ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Finite Group Actions ◽  
Closed Riemann Surface ◽  
A Finite Group ◽  
The Given

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fgbounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

THE STRONG SYMMETRIC GENUS OF DIRECT PRODUCTS

2011 ◽  
Vol 10 (05) ◽  
pp. 901-914 ◽  
Author(s):  
COY L. MAY
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Direct Product ◽  
Direct Products ◽  
A Finite Group

Let G be a finite group. The strong symmetric genus σ0(G) is the minimum genus of any Riemann surface on which G acts faithfully and preserving orientation. Assume that G is non-abelian and generated by two elements, one of which is an involution, and that n is relatively prime to |G|. Our first main result is the determination of the strong symmetric genus of the direct product Zn ×G in terms of n, |G|, and a parameter associated with the group G. We obtain a variety of genus formulas. Finally, we apply these results to prove that for each integer g ≥ 2, there are at least four groups of strong symmetric genus g.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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