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- W2883706064 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 $$begin{aligned} {bar{H}}^2(mathbb {Z}/n) = begin{array}{c} lim {mathop {scriptstyle varGamma }limits ^{textstyle rightarrow }} end{array} H^2(varGamma ,mathbb {Z}/n), end{aligned}$$ where $$varGamma $$ 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 of $${bar{H}}^2(mathbb {Z}/n)$$ 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 $$begin{aligned} {bar{H}}^2({mathbb {Z}_l}) = begin{array}{c} lim {mathop {scriptstyle t}limits ^{textstyle leftarrow }} end{array} {bar{H}}^2(mathbb {Z}/l^t). end{aligned}$$ 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.
- W2883706064 created "2018-08-03" @default.
- W2883706064 creator A5003689747 @default.
- W2883706064 date "2018-11-02" @default.
- W2883706064 modified "2023-10-16" @default.
- W2883706064 title "Non-residually finite extensions of arithmetic groups" @default.
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- W2883706064 doi "https://doi.org/10.1007/s40993-018-0140-z" @default.
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