Non-residually finite extensions of arithmetic groups

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 $b+m$.