On the Galois structure of the class group of certain Kummer extensions

AbstractChinburg's third invariant Ω(N/K, 3) ∊ C1(Z[Γ]) of a Galois extension N/K of number fields with group Γ is closely related to the Galois structure of unit groups and ideal class groups, and deep unsolved problems such as Stark's conjecture.We give a formula for Ω(N/K, 3) modulo D(ZΓ) in the factorisability defect class group, reminiscent of analytic class number formulas. Specialising to the case of an absolutely abelian, real field N, we give a natural conjecture in terms of Hecke factorisations which implies the vanishing of the invariant in the defect class group.We prove this conjecture when N has prime-power conductor using Euler systems of cyclotomic units, Ramachandra units and Hecke factorisation. This supports a general conjecture of Chinburg, which in our situation specialises to the statement that Ω(N/K, 3) = 0 for such extensions.We also develop a slightly extended version of Euler systems of units for general abelian extensions, which will be applied to abelian extensions of imaginary quadratic fields elsewhere

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Generating the Mapping Class Group

10.23943/princeton/9780691147949.003.0005 ◽

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.

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A Primer on Mapping Class Groups (PMS-49)

10.23943/princeton/9780691147949.001.0001 ◽

2017 ◽

Cited By ~ 1

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.

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Electoral Integrity in America

10.1093/oso/9780190934163.001.0001 ◽

2018 ◽

Cited By ~ 9

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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Team formation methods for increasing interaction during in-class group work

ACM SIGCSE Bulletin ◽

10.1145/1151954.1067525 ◽

2005 ◽

Vol 37 (3) ◽

pp. 291-295 ◽

Cited By ~ 14

Author(s):

Katherine Deibel

Keyword(s):

Group Work ◽

Class Group ◽

Team Formation

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Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

Journal of Algebraic Combinatorics ◽

10.1007/s10801-021-01025-x ◽

2021 ◽

Author(s):

Michele Rossi ◽

Lea Terracini

Keyword(s):

Free Group ◽

Toric Variety ◽

Class Group ◽

Abelian Groups ◽

Toric Varieties ◽

Picard Group ◽

Closed Field ◽

Finitely Generated ◽

Free Part ◽

Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

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The local-to-global property for Morse quasi-geodesics

Mathematische Zeitschrift ◽

10.1007/s00209-021-02811-w ◽

2021 ◽

Author(s):

Jacob Russell ◽

Davide Spriano ◽

Hung Cong Tran

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Type Theorem ◽

Hyperbolic Groups ◽

Global Property ◽

Mapping Class ◽

Relatively Hyperbolic Groups ◽

Convex Cocompact ◽

Relatively Hyperbolic ◽

Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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