Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 6 (2011), 161 -- 164

A NOTE OF ZUK'S CRITERION

Traian Preda

Abstract. Zuk's criterion give us a condition for a finitely generated group to have Property (T): the smallest non - zero eigenvalue of Laplace operator Δ_μ corresponding to the simple random walk on G(S) satisfies λ₁(G) > 1/2. We present here two examples that prove that this condition cannot be improved.

2010 Mathematics Subject Classification: 22D10.
Keywords: Property (T); Zuk's criterion; Spectrum of the Laplace operator.

Full text

References

1. B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), Monography, Cambridge University Press, 2008. MR2415834.

2. M. Gromov, Random walks in random groups, Geom. Funct. Anal. (GAFA), 13 (2003), 73-146. MR1978492. Zbl 1122.20021.

3. A. Zuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. (GAFA), 13 (2003), 643-670. MR1995802(2004m:20079). Zbl 1036.22004.

Traian Preda
University of Bucharest,
Str. Academiei nr.14, Bucureşti,
România.
e-mail: traianpr@yahoo.com

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