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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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