Non-residually finite extensions of arithmetic groups

TL;DR

This paper demonstrates that many finite extensions of arithmetic groups are not residually finite by analyzing their cohomology groups and central extensions, revealing new non-residually finite structures.

Contribution

It introduces a cohomological framework to identify and classify non-residually finite extensions of arithmetic groups, including their invariance properties and lifting conditions.

Findings

Infinite elements of order n in cohomology groups indicate non-residually finite extensions.

Invariant elements under the arithmetic completion correspond to extensions that lift to characteristic zero.

The dimension c of the invariant cohomology group relates to the structure of the real points of the algebraic group.

Abstract

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose $G$ is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of $G$ has finite extensions which are not residually finite. More precisely, we investigate the group \[ \bar H^2(\mathbb{Z}/n) = direct limit ( H^2(\Gamma,\mathbb{Z}/n) ), \] where $\Gamma$ runs through the arithmetic subgroups of $G$. Elements of $\bar H^2(\mathbb{Z}/n)$ correspond to (equivalence classes of) central extensions of arithmetic groups by $\mathbb{Z}/n$; non-zero elements correspond to extensions which are not residually finite. We prove that $\bar H^2(\mathbb{Z}/n)$ contains infinitely many elements of order $n$, some of which are invariant for the action of the arithmetic completion $\widehat{G(\mathbb{Q})}$ of $G(\mathbb{Q})$. We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group \[ \bar H^2(\mathbb{Z}_l) = projective limit \bar H^2(\mathbb{Z}/l^t). \] We show that $\bar H^2(\mathbb{Z}_l)^{\widehat{G(\mathbb{Q})}}$ is isomorphic to $\mathbb{Z}_l^c$ for some positive integer $c$. When $G(\mathbb{R})$ has no simple components of complex type, we prove that $c=b+m$, where $b$ is the number of simple components of $G(\mathbb{R})$ and $m$ is the dimension of the centre of a maximal compact subgroup of $G(\mathbb{R})$. In all other cases, we prove upper and lower bounds on $c$; our lower bound (which we believe is the correct number) is $b+m$.