Non-residually finite extensions of arithmetic groups View Full Text


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Article Info

DATE

2019-03

AUTHORS

Richard M. Hill

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),where Γ runs through the arithmetic subgroups of G. Elements of H¯2(Z/n) correspond to (equivalence classes of) central extensions of arithmetic groups by Z/n; non-zero elements of H¯2(Z/n) correspond to extensions which are not residually finite. We prove that H¯2(Z/n) contains infinitely many elements of order n, some of which are invariant for the action of the arithmetic completion G(Q)^ of G(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).We show that H¯2(Zl)G(Q)^ is isomorphic to Zlc for some positive integer c. When G(R) has no simple components of complex type, we prove that c=b+m, where b is the number of simple components of G(R) and m is the dimension of the centre of a maximal compact subgroup of 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. More... »

PAGES

2

References to SciGraph publications

  • 2002-09. The residual finiteness of negatively curved polygons of finite groups in INVENTIONES MATHEMATICAE
  • 1965-12. Introduction in PUBLICATIONS MATHÉMATIQUES DE L'IHÉS
  • 1994. Cohomologie Galoisienne, Cinquième édition, révisée et complétée in NONE
  • 1973-12. Corners and arithmetic groups in COMMENTARII MATHEMATICI HELVETICI
  • 1974-09. Continuous cohomology and a conjecture of Serre's in INVENTIONES MATHEMATICAE
  • 1994-12. Cohomology ofS-arithmetic subgroups in the number field case in INVENTIONES MATHEMATICAE
  • Journal

    TITLE

    Research in Number Theory

    ISSUE

    1

    VOLUME

    5

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s40993-018-0140-z

    DOI

    http://dx.doi.org/10.1007/s40993-018-0140-z

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1107989575


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