A topological realization of the congruence subgroup Kernel A

TL;DR

This paper reveals a topological interpretation of the congruence subgroup kernel for SL(2) by relating it to the fundamental group of a certain covering space derived from compactifications of locally symmetric spaces.

Contribution

It establishes a novel topological realization of the congruence subgroup kernel using inverse limits of compactifications, connecting algebraic and topological perspectives.

Findings

The congruence subgroup kernel appears as the deck transformation group of a specific covering space.

A construction from inverse limits of reductive Borel-Serre compactifications is used.

The fundamental group computation from previous work is key to this realization.

Abstract

A number of years ago, Kumar Murty pointed out to me that the computation of the fundamental group of a Hilbert modular surface ([7],IV,${\S}$6), and the computation of the congruence subgroup kernel of SL(2) ([6]) were surprisingly similar. We puzzled over this, in particular over the role of elementary matrices in both computations. We formulated a very general result on the fundamental group of a Satake compactification of a locally symmetric space. This lead to our joint paper [1] with Lizhen Ji and Les Saper on these fundamental groups. Although the results in it were intriguingly similar to the corresponding calculations of the congruence subgroup kernel of the underlying algebraic group in [5], we were not able to demonstrate a direct connection (cf. [1], ${\S}$7). The purpose of this note is to explain such a connection. A covering space is constructed from inverse limits of reductive Borel-Serre compactifications. The congruence subgroup kernel then appears as the group of deck transformations of this covering. The key to this is the computation of the fundamental group in [1].