## Algebra & Number Theory

### The congruence topology, Grothendieck duality and thin groups

#### Abstract

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

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

Dates
Revised: 20 January 2019
Accepted: 8 March 2019
First available in Project Euclid: 21 August 2019

https://projecteuclid.org/euclid.ant/1566353008

Digital Object Identifier
doi:10.2140/ant.2019.13.1281

Mathematical Reviews number (MathSciNet)
MR3994565

Zentralblatt MATH identifier
07103974

Keywords
congruence subgroup thin groups

#### Citation

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. https://projecteuclid.org/euclid.ant/1566353008

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