Units of classical orders: A survey
A valuable account on theorems and methods for the investigation of unit groups of orders in semisimple \(\mathbb{Q}\)-algebras \(A\) is given. Having in mind Dirichlet's unit theorem as the most classical fact in this area, the author presents a number of generalizations and structural results on unit groups which enable him, at the end of this collection, to speculate on a ``general unit theorem which would have satisfied Eichler''. In the course of his discussion, he considers three different generalizations of Dirichlet's theorem: Firstly, for an order in a division algebra \(D\), the unit group \(\Gamma\) appears as a cocompact subgroup of the unit group of \(\mathbb{R}\otimes D\), which implies (the hard part of) Dirichlet's unit theorem. The generalization of cocompactness to arithmetic groups \(\Gamma\) suggests itself. In this context, finite presentation of \(\Gamma\) is discussed. Weyl's treatment on fundamental domains for lattice groups in skewfields is explained in some detail. Swan's paper on \(\text{SL}_2(R)\) is mentioned, where \(R\) denotes the integers in an imaginary quadratic field, and Frohman and Fine's explicit description of \(\text{SL}_2(R)\) as a free product with amalgamation. Secondly, a formula for the virtual cohomological dimension of a unit group \(\Gamma\) is given which yields another generalization of Dirichlet's unit theorem. In particular, it is proved that \(\text{SL}_2(\mathbb{Z})\) is the unique virtually free unit group. A third generalization of Dirichlet's theorem is obtained from a unit theorem of \textit{H. Bass} [Topology 4, 391-410 (1966; Zbl 0166.02401)], involving the Whitehead groups of the given order. Several other topics are discussed in the paper, e.g., the congruence subgroup problem, Eichler's treatment of indefinite quaternion orders and their unit groups, the relationship between \(K_2(R)\) and presentations of \(\text{SL}_n(R)\) for \(n\geq 3\), etc.
scientific article; zbMATH DE number 856840