Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U\_n(\mathbb{C})$. The category of these representations is an exact category, and so we can form the Grothendieck group $K\_0(G)$ of this category. If $H$ is a finite-index subgroup of $G$, we have an induction map
$K\_0(H) \rightarrow K\_0(G)$. Given a collection of subgroups $H\_i$ of $G$, we can ask whether the map obtained by induction


$\oplus\_i K\_0(H\_i) \rightarrow K\_0(G)$


is onto, or onto after tensoring with $\mathbb{Q}$. Let us say that the collection is good if this map is onto. If $G$ is finite, for example the Brauer induction theorem or the Artin induction theorem give answers: The collection of elementary subgroups of $G$ is good. If $G$ is infinite and maps to a finite group $F$, we can pull back the elementary subgroups of $F$ to $G$ which yields a good collection of subgroups of $G$.


My question is: Are there other known cases of infinite groups $G$ with a good collection of subgroups? Also, I would be interested in any references where finite-dimensional representations of infinite groups are studied.



Induction theorems for finite-dimensional complex representations of infinite groups
I believe that all examples must come from finite quotients:


1. A collection of subgroups is necessarily good if the element $1\in K\_0(G)$ is in the subgroup generated by the images of the induction maps $K\_0(H)\to K\_0(G)$. That's because these images are ideals, using the formula $V\otimes Ind(W)=Ind(Res(V)\otimes W)$.
2. Therefore every good collection contains a finite good collection. And of course for any collection of finite index subgroups $H\_i$ there is a finite index normal subgroup $N$ of $G$ that is contained in them all.
3. And then the collection ${H\_i}$ will be good in $G$ if and only if the collection $H\_i/N$ is good in the finite group $G/N$. (This last requires a little argument splitting representations of $G$, or $H\_i$, into the part fixed by $N$ and its orthogonal complement and noting that this splitting is compatible with induction.)


