From Farey Symbols to Generators for Subgroups of Finite Index in Integral Group Rings of Finite Groups

Ann Dooms, Eric Jespers, A. Konovalov

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

Let U(ZG) be the group of units of the integral group ring ZG, of the finite
nonabelian group G over the ring of integers Z. Although work of Borel and
Harish-Chandra showed almost a half century ago that U(ZG) is finitely pre-
sented, a specific set of generators for the group is known only in a few cases.
Thus the focus of research has been finding specific generators for subgroups of
finite index in U(ZG). The current paper extends work of many authors over
a considerable period of time on this topic. The principle achievement of the
paper is to eliminate the assumption that the rational group ring QG does not
contain simple exceptional Wedderburn components of the type M2(Q). The
problems caused by such factors are well known to experts on this topic. The
paper overcomes these problems by introducing new units, using the theory of
Farey symbols that yield generators for subgroups of finite index in PSL2(Z).
As could be expected, Poincare's method which represents PSL2(Z) as Moebius
transformations in the hyperbolic plane plays an important role in the above.
Original languageEnglish
Pages (from-to)263-283
JournalJournal of K-Theory
Volume6
Issue number2
Publication statusPublished - 2010

Keywords

  • unit group

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