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Thin groups and superstrong approximation / [edited by] Emmanuel Breuillard and Hee Oh.

By: Contributor(s): Material type: TextTextSeries: Mathematical Sciences Research Institute publications ; 61.Publication details: New York: Cambridge University Press, 2014.Description: xii, 362 p. : illustrations ; 24 cmISBN:
  • 9781107036857 (hardback : alk. paper)
Subject(s): DDC classification:
  • 512.482 23 W926
Contents:
Some Diophantine applications of the theory of group expansion -- A brief introduction to approximate groups -- Superstrong approximation for monodromy groups -- The ubiquity of thin groups -- The orbital circle method -- Sieves in discrete groups, expecially sparse -- How random are word maps? -- Constructing thin groups -- Ergodic properties of the Burger-Roblin measure -- Harmonic analysis, ergodic theory and counting for thin groups -- Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces -- Growth in linear groups -- Strong approximation for algebraic groups -- Generic phenomena in groups: some answers and many questions -- Affine sieve and expanders -- Notes on thin matrix groups.
Summary: This is a collection of surveys and primarily expository articles focusing on recent developments concerning various quantitative aspects of 'thin groups.' There are discrete subgroups of semisimple Lie groups that are both big (Zariski dense) and small (of infinite covolume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading, for instance, to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap, referred to a superstrong aproximation. This book provides a broad panorama of a very active field of mathematics at the boundary between geometry, dynamical systems, number theory, and combinatorics.
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Includes bibliographical references and index.

Some Diophantine applications of the theory of group expansion --
A brief introduction to approximate groups --
Superstrong approximation for monodromy groups --
The ubiquity of thin groups --
The orbital circle method --
Sieves in discrete groups, expecially sparse --
How random are word maps? --
Constructing thin groups --
Ergodic properties of the Burger-Roblin measure --
Harmonic analysis, ergodic theory and counting for thin groups --
Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces --
Growth in linear groups --
Strong approximation for algebraic groups --
Generic phenomena in groups: some answers and many questions --
Affine sieve and expanders --
Notes on thin matrix groups.

This is a collection of surveys and primarily expository articles focusing on recent developments concerning various quantitative aspects of 'thin groups.' There are discrete subgroups of semisimple Lie groups that are both big (Zariski dense) and small (of infinite covolume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading, for instance, to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap, referred to a superstrong aproximation. This book provides a broad panorama of a very active field of mathematics at the boundary between geometry, dynamical systems, number theory, and combinatorics.

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