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Linear representations of binate groups. (English) Zbl 0813.20060
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.

MSC:
20J05 Homological methods in group theory
20F05 Generators, relations, and presentations of groups
20C15 Ordinary representations and characters
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