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- W4289743064 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 $b+m$." @default.
- W4289743064 created "2022-08-04" @default.
- W4289743064 creator A5040242777 @default.
- W4289743064 date "2018-07-30" @default.
- W4289743064 modified "2023-10-17" @default.
- W4289743064 title "Non-residually finite extensions of arithmetic groups" @default.
- W4289743064 doi "https://doi.org/10.48550/arxiv.1807.11449" @default.
- W4289743064 hasPublicationYear "2018" @default.
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