Linear representations of binate groups
A binate group \(G\) is a group with the property that for each finitely generated, proper subgroup \(H < G\), there is a homomorphism \(\varphi : H \to G\), and an element \(t \in G \setminus H\) such that for all \(h \in H\), one has \(h = [t,\varphi(h)]\). Such a group is perfect and acyclic (i.e. has vanishing integral homology in all positive dimensions). It is first shown using homological arguments that binate groups have no non-trivial finite-dimensional unitary representations, and then, independently, using results about algebraic groups, that in fact they have no non-trivial finite-dimensional linear representations whatsoever.
scientific article; zbMATH DE number 609788
scientific article