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The proalgebraic completion of rigid groups. (English) Zbl 1059.20036
Summary: A finitely generated group \(\Gamma\) is called ‘representation rigid’ (briefly, rigid) if for every \(n\), \(\Gamma\) has only finitely many classes of simple \(\mathbb{C}\) representations in dimension \(n\). Examples include higher rank \(S\)-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are ‘representation super rigid’; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.

20F65 Geometric group theory
20C15 Ordinary representations and characters
20E18 Limits, profinite groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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