From the Poincaré theorem to generators of the unit group of integral group rings of finite groups. (English) Zbl 1326.16034
Let \(\mathcal O\) be a \(\mathbb Z\)-order in a finite dimensional semisimple \(\mathbb Q\)-algebra. The determination of the unit group \(\mathcal O^\times\) of \(\mathcal O\), a finitely presented group, is a notoriously difficult problem. For a group ring \(\mathcal O\) over an abelian group, H. Bass [Topology 4, 391-410 (1966; Zbl 0166.02401)] proved a generalized Dirichlet unit theorem which implies that the “Bass units” generate a subgroup of finite index in \(\mathcal O^\times\). Refinements of this theorem have been obtained by Ritter and Segal, and E. Jespers and G. Leal [Manuscr. Math. 86, No. 4, 479-498 (1995; Zbl 0834.16034)]. The latter result works for a large class of finite groups, with the exception of those which admit a non-abelian fixedpoint-free epimorphic image. Moreover, some exceptional components of the rational group algebra must be excluded. Then it is shown that the Bass units and so-called bicyclic units generate a subgroup of finite index.
The paper under review provides a general method to handle two types of exceptional simple components, namely, \(2\times 2\) matrix algebras over fields as well as quaternion algebras over \(\mathbb Q\) or over an imaginary quadratic field. The results are obtained via actions on hyperbolic spaces. This leads to very effective algorithms which calculate generating systems for subgroups of finite index.
The paper under review provides a general method to handle two types of exceptional simple components, namely, \(2\times 2\) matrix algebras over fields as well as quaternion algebras over \(\mathbb Q\) or over an imaginary quadratic field. The results are obtained via actions on hyperbolic spaces. This leads to very effective algorithms which calculate generating systems for subgroups of finite index.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 16U60 | Units, groups of units (associative rings and algebras) |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 16S34 | Group rings |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
groups of units; integral group rings; fundamental domains; generators; subgroups of finite index; bicyclic units; Bass unitsSoftware:
CongruenceReferences:
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