Algebra & Number Theory

The congruence topology, Grothendieck duality and thin groups

Alexander Lubotzky and Tyakal Nanjundiah Venkataramana

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This paper answers a question raised by Grothendieck in 1970 on the “Grothendieck closure” of an integral linear group and proves a conjecture of the first author made in 1980. This is done by a detailed study of the congruence topology of arithmetic groups, obtaining along the way, an arithmetic analogue of a classical result of Chevalley for complex algebraic groups. As an application we also deduce a group theoretic characterization of thin subgroups of arithmetic groups.

Article information

Algebra Number Theory, Volume 13, Number 6 (2019), 1281-1298.

Received: 8 May 2018
Revised: 20 January 2019
Accepted: 8 March 2019
First available in Project Euclid: 21 August 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E57: Classical groups [See also 14Lxx, 20Gxx]
Secondary: 20G30: Linear algebraic groups over global fields and their integers

congruence subgroup thin groups


Lubotzky, Alexander; Venkataramana, Tyakal Nanjundiah. The congruence topology, Grothendieck duality and thin groups. Algebra Number Theory 13 (2019), no. 6, 1281--1298. doi:10.2140/ant.2019.13.1281.

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