## 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.

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 language | English |
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Pages (from-to) | 263-283 |

Journal | Journal of K-Theory |

Volume | 6 |

Issue number | 2 |

Publication status | Published - 2010 |

## Keywords

- unit group