Virtual Cohomological Dimension of Unit Groups

Definitions and mathematical statement

For a finite group $G$, let \[ {\mathbb{Q}} G\cong A_1\times\cdots\times A_n \] be its Wedderburn decomposition into simple rational algebras. Write $K_i=Z(A_i)$ for the center. An order $R_i\subset A_i$ is a subring with identity that is a full finitely generated ${\mathbb{Z}}$-lattice in $A_i$. Its reduced-norm-one unit group is \[ R_i^1=\{u\in R_i^\times:\operatorname{Nrd}_{A_i/K_i}(u)=1\}. \] For a virtually torsion-free group $H$, its virtual cohomological dimension is $\operatorname{vcd}(H)=\operatorname{cd}_{{\mathbb{Z}}}(H_0)$ for a torsion-free subgroup $H_0$ of finite index; it is independent of that choice.

The source reports classifications when \[ \operatorname{vcd}(R_i^1)\leq2\qquad\text{for every }i. \] The requested extension is to the class of finite groups and associated rational components satisfying \[ \operatorname{vcd}(R_i^1)\leq3\qquad\text{for every }i, \] and to obtain a corresponding classification or structural description of \[ U({\mathbb{Z}} G)=({\mathbb{Z}} G)^\times. \] The condition is componentwise; it is not replaced by the different requirement $\operatorname{vcd}(U({\mathbb{Z}} G))\leq3$. The full integral unit group is the target, not only its augmentation-one subgroup.

The extract does not state the existing classification theorems, the intended output (a list of finite groups, allowed simple components, or descriptions up to a specified group equivalence), or whether the unit-group classification is exact or up to finite index/commensurability. No direct-product identity for the integral unit group is inferred merely from the rational Wedderburn product, and the dimension cutoff alone is not presented as a classification theorem.

Short English statement

Extend the known componentwise norm-one-unit classifications from virtual cohomological dimension at most two to at most three, and describe the resulting integral group-ring unit groups. The intended classification equivalence and baseline theorems are not supplied.

Sources

• Supplied report: Mini-Workshop: Arithmetik von Gruppenringen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/55, pp. 3209--3240; proposed year 2007. https://doi.org/10.4171/owr/2007/55.
• Local UnsolvedMath registry, permanent ID 30000885, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the complete extension question and indexed objects were restored. The supplied 21 August 2026 literature triage labels the general extension open, reporting restricted families rather than a full classification. No additional external reference is supplied; this dated assessment is not independently verified here.