The congruence topology, Grothendieck duality and thin groups

TL;DR

This paper explores the congruence topology of arithmetic groups, addressing Grothendieck's 1970 question and a 1980 conjecture, leading to new insights into thin subgroups and an arithmetic analogue of Chevalley's theorem.

Contribution

It provides a detailed study of the congruence topology, proving Grothendieck's closure question and a conjecture, and characterizes thin subgroups within arithmetic groups.

Findings

Solved Grothendieck's 1970 question on the closure of integral linear groups.

Proved a conjecture from 1980 regarding the structure of these groups.

Developed an arithmetic analogue of Chevalley's classical result.

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.