Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Boundaries, Weyl groups, and Superrigidity

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

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Uri Bader - Mathematics Department, The Technion, 32000 Haifa, Israel (email)
Alex Furman - Mathematics, Statistics and Computer Science Department, University of Illinois at Chicago, Chicago, 851 S. Morgan St., Illinois, 60607, United States (email)

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.

Keywords:  Weyl groups, superrigidity, lattices.
Mathematics Subject Classification:  Primary: 20E40.

Received: September 2011;      Available Online: March 2012.