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 H¯2(Z/n)=limΓ→H2(Γ,Z/n),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2(\\mathbb {Z}/n) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle \\varGamma }\\limits ^{\\textstyle \\rightarrow }} \\end{array} H^2(\\varGamma ,\\mathbb {Z}/n), \\end{aligned}$$\\end{document}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 H¯2(Zl)=limt←H¯2(Z/lt).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2({\\mathbb {Z}_l}) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle t}\\limits ^{\\textstyle \\leftarrow }} \\end{array} {\\bar{H}}^2(\\mathbb {Z}/l^t). \\end{aligned}$$\\end{document}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.