The congruence topology, Grothendieck duality and thin groups

Lubotzky, Alexander; Venkataramana, Tyakal Nanjundiah

doi:10.2140/ant.2019.13.1281

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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.

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

Vol. 13, No. 6, 2019