Boundaries, Weyl groups, and Superrigidity

Uri Bader; Alex Furman

doi:10.3934/era.2012.19.41

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2012, Volume 19: 41-48. Doi: 10.3934/era.2012.19.41

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Boundaries, Weyl groups, and Superrigidity

Uri Bader^1, and
Alex Furman^2,

1.
Mathematics Department, The Technion, 32000 Haifa
2.
Mathematics, Statistics and Computer Science Department, University of Illinois at Chicago, Chicago, 851 S. Morgan St., Illinois, 60607

Received: August 31, 2011

Published: January 2012

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Abstract

This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $\Gamma$ and a boundary action $\Gamma$ ↷ $B$ we associate a certain generalized Weyl group $W_{{\Gamma}{B}}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \bf{H}$, defines a special homomorphism $W_{{\Gamma}{B}}\to Weyl_{\bf H}$. This general fact allows the deduction of the aforementioned superrigidity results.

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Mathematics Subject Classification: Primary: 20E40.

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