The proalgebraic completion of rigid groups

Hyman Bass; Alexander Lubotzky; Andy R. Magid; Shahar Mozes

doi:10.1023/A:1021221727311

The proalgebraic completion of rigid groups

Hyman Bass^*, Alexander Lubotzky, Andy R. Magid, Shahar Mozes

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

24 Scopus citations

Abstract

A finitely generated group Γ is called representation rigid (briefly, rigid) if for every n, Γ has only finitely many classes of simple 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.

Original language	English
Pages (from-to)	19-58
Number of pages	40
Journal	Geometriae Dedicata
Volume	95
Issue number	1
DOIs	https://doi.org/10.1023/A:1021221727311
State	Published - 2002

Bibliographical note

Funding Information:
The authors collectively, individually, and in various subsets, gratefully acknowledge support at various times from the United States Israel Binational Science Foundation, the Israel Science Foundation, the United States National Security Agency and National Science Foundation, Hebrew University, the University of Oklahoma, Columbia University and the University of Michigan.

Keywords

Finitely generated group
Linear representation

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10.1023/A:1021221727311

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AB - A finitely generated group Γ is called representation rigid (briefly, rigid) if for every n, Γ has only finitely many classes of simple 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.

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The proalgebraic completion of rigid groups

Abstract

Bibliographical note

Keywords

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