Abstract
We prove that the universal lattices – the groups G=SLd(R) where R=ℤ[x 1,...,x k], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K 2 of some finite commutative rings and a relative property T of \((\mathrm{SL}_2(R)\ltimes R^2, R^2)\).
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Mathematics Subject Classification (2000)
20F69, 13M05, 19C20, 20G05, 20H05
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Kassabov, M., Nikolov, N. Universal lattices and property tau. Invent. math. 165, 209–224 (2006). https://doi.org/10.1007/s00222-005-0498-0
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DOI: https://doi.org/10.1007/s00222-005-0498-0