Representatives of the Connected Component of the Idèle Class Group

1965 ◽  
pp. 249-252 ◽  
Author(s):  
Emil Artin
Keyword(s):  
Class Group ◽  
Connected Component

scholarly journals Curves with prescribed symmetry and associated representations of mapping class groups

2021 ◽  
Author(s):  
Marco Boggi ◽  
Eduard Looijenga
Keyword(s):  
Algebraic Curve ◽  
Class Group ◽  
Mapping Class ◽  
Zariski Closure ◽  
Topological Properties ◽  
Connected Component ◽  
Class Groups ◽  
Almost Simple Group ◽  
Linear Representations ◽  
A Finite Group

AbstractLet C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra $${{\mathbb {Q}}}G$$ Q G to the algebra of $${{\mathbb {Q}}}$$ Q -endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation often acts $${{\mathbb {Q}}}$$ Q -irreducibly in a G-isogeny space of $$H^1(C; {{\mathbb {Q}}})$$ H 1 ( C ; Q ) and with image a $${{\mathbb {Q}}}$$ Q -almost simple group.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Electoral Integrity in America

2018 ◽  
Keyword(s):  
News Media ◽  
Class Group ◽  
Public Trust ◽  
Red Flags ◽  
Fake News ◽  
Electoral Fraud ◽  
State Regulations ◽  
Electoral Laws ◽  
American Elections

The contemporary era raises a series of red flags about electoral integrity in America. Problems include plummeting public trust, exacerbated by President Trump’s claims of massive electoral fraud. Confidence in the impartiality and reliability of information from the news media has eroded. And Russian meddling has astutely exploited both these vulnerabilities, heightening fears that the 2016 contest was unfair. This book brings together a first-class group of expert academics and practitioners to analyze challenges facing contemporary elections in America. Contributors analyze evidence for a series of contemporary challenges facing American elections, including the weaknesses of electoral laws, overly restrictive electoral registers, gerrymandering district boundaries, fake news, the lack of transparency, and the hodgepodge of inconsistent state regulations. The conclusion sets these issues in comparative context and draws out the broader policy lessons for improving electoral integrity and strengthening democracy.

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