, Volume 179, Issue 2, pp 303-347
Date: 25 Sep 2009

Property (T) for noncommutative universal lattices

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Abstract

We establish a new spectral criterion for Kazhdan’s property (T) which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property (T) for the groups EL n (R), where n≥3 and R is an arbitrary finitely generated associative ring. We also strengthen some of the results on property (T) for Kac-Moody groups from (Dymara and Januszkiewicz in Invent. Math. 150(3):579–627, 2002).

The second author is supported by the Spanish Ministry of Science and Innovation, the grant MTM2008-06680 and the Autonomous University of Madrid, the grant CCG08-UAM/ESP-4145. This project was initiated during the second author’s visit to University of Virginia in March, 2008. The authors thank University of Virginia for its hospitality.