Vol. 13, No. 6, 2019

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The congruence topology, Grothendieck duality and thin groups

Alexander Lubotzky and Tyakal Nanjundiah Venkataramana

Vol. 13 (2019), No. 6, 1281–1298
DOI: 10.2140/ant.2019.13.1281
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.

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Keywords
congruence subgroup, thin groups
Mathematical Subject Classification 2010
Primary: 11E57
Secondary: 20G30
Milestones
Received: 8 May 2018
Revised: 20 January 2019
Accepted: 8 March 2019
Published: 18 August 2019
Authors
Alexander Lubotzky
Einstein Institute of Mathematics
The Hebrew University of Jerusalem
Israel
Tyakal Nanjundiah Venkataramana
School of Mathematics
Tata Institute of Fundamental Research
Mumbai
India