On the Galois structure of arithmetic cohomology II: ray class groups

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

Download Full-text

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0034 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Jiuya Wang

Keyword(s):

Galois Group ◽

Prime Number ◽

Finite Groups ◽

Upper Bound ◽

Abelian Groups ◽

Class Groups ◽

Abelian Extensions ◽

Pointwise Bound ◽

First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

Download Full-text

Galois structure of Zariski cohomology for weakly ramified covers of curves

American Journal of Mathematics ◽

10.1353/ajm.2004.0037 ◽

2004 ◽

Vol 126 (5) ◽

pp. 1085-1107 ◽

Cited By ~ 10

Author(s):

Bernhard Kock

Keyword(s):

Galois Structure ◽

Ramified Covers

Download Full-text

Social Class and Code Elaboration in Oral Communication

Journal of Speech and Hearing Research ◽

10.1044/jshr.1402.421 ◽

1971 ◽

Vol 14 (2) ◽

pp. 421-427 ◽

Cited By ~ 5

Author(s):

Millicent E. Poole ◽

T. W. Field

Keyword(s):

Social Class ◽

Middle Class ◽

Working Class ◽

Oral Communication ◽

Class Groups ◽

Parental Occupation ◽

The Social ◽

Linguistic Coding

The Bernstein thesis of elaborated and restricted coding orientation in oral communication was explored at an Australian tertiary institute. A working-class/middle-class dichotomy was established on the basis of parental occupation and education, and differences in overall coding orientation were found to be associated with social class. This study differed from others in the area in that the social class groups were contrasted in the totality of their coding orientation on the elaborated/restricted continuum, rather than on discrete indices of linguistic coding.

Download Full-text