Non-residually finite extensions of arithmetic groups

摘要

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 $$egin{aligned} {ar{H}}^2(mathbb {Z}) = egin{array}{c} lim {mathop {scriptstyle arGamma }limits ^{extstyle ightarrow }} end{array} H^2(arGamma ,mathbb {Z}), end{aligned}$$ H ˉ 2 ( Z / n ) = lim Γ → H 2 ( Γ , Z / n ) , where $$arGamma $$ Γ runs through the arithmetic subgroups of G . Elements of $${ar{H}}^2(mathbb {Z})$$ H ˉ 2 ( Z / n ) correspond to (equivalence classes of) central extensions of arithmetic groups by $$mathbb {Z}$$ Z / n ; non-zero elements of $${ar{H}}^2(mathbb {Z})$$ H ˉ 2 ( Z / n ) correspond to extensions which are not residually finite. We prove that $${ar{H}}^2(mathbb {Z})$$ H ˉ 2 ( Z / n ) contains infinitely many elements of order n , some of which are invariant for the action of the arithmetic completion $${widehat{G(mathbb {Q})}}$$ G ( Q ) ^ of $$G(mathbb {Q})$$ G ( Q ) . We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group $$egin{aligned} {ar{H}}^2({mathbb {Z}_l}) = egin{array}{c} lim {mathop {scriptstyle t}limits ^{extstyle leftarrow }} end{array} {ar{H}}^2(mathbb {Z}/l^t). end{aligned}$$ H ˉ 2 ( Z l ) = lim t ← H ˉ 2 ( Z / l t ) . We show that $${ar{H}}^2({mathbb {Z}_l})^{widehat{G(mathbb {Q})}}$$ H ˉ 2 ( Z l ) G ( Q ) ^ is isomorphic to $${mathbb {Z}_l}^c$$ Z l c for some positive integer c . When $$G(mathbb {R})$$ G ( R ) has no simple components of complex type, we prove that $$c=b+m$$ c = b + m , where b is the number of simple components of $$G(mathbb {R})$$ G ( R ) and m is the dimension of the centre of a maximal compact subgroup of $$G(mathbb {R})$$ G ( R ) . In all other cases, we prove upper and lower bounds on c ; our lower bound (which we believe is the correct number) is $$b+m$$ b + m .