Surveys in Mathematics and its Applications
ISSN1842-6298 (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 ∆µcorresponding to the simple random walk onG(S) satisfies λ1(G) > 12. We present here two examples that prove that this condition cannot be improved.
Definition 1. (see [1] and [2] )
i) A random walk or Markov kernel on a non-empty set X is a kernel with non- negative values µ:X×X →R+ such that:
X
y∈X
µ(x, y) = 1,∀x∈X.
ii) A stationary measure for a random walk µ is a function ν :X → R∗+ such that :
ν(x)µ(x, y) =ν(y)µ(y, x),∀x, y∈X.
Example 2. Let G =(X,E) be a locally finite graph. For x,y ∈ X, set
µ(x, y) =
1
deg(x) if(x, y)∈E
0 otherwise
(0.1) and deg(x) =card {y∈X|(x, y)∈E} is the degree of a vertex x ∈ X.
µ is called simple random walk on X and ν is a stationary measure for µ.
Consider the Hilbert space:
Ω0C(X) ={f :X →C|X
x∈X
|f(x)|2ν(x)<∞}
The Laplace operator ∆µon Ω0
C(X) is definited by (∆µf)(x) =f(x)−X
x∼y
f(y)µ(x, y).
2010 Mathematics Subject Classification: 22D10.
Keywords: Property (T); Zuk’s criterion; Spectrum of the Laplace operator.
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http://www.utgjiu.ro/math/sma
162 T. Preda Let Γ be a group generated by a finite set S. We assume that e∈/ S and S = S−1 (S is simetric).
The graph G(S) associated toS has vertex set S and the set of edges is the set of pairs (s, t)∈S×S such thats−1t∈S.
Theorem 3. (Zuk’s criterion)( see [3])
Let Γ be a group generated by a finite set S with e /∈S. Let G(S) be the graph associated to S. Assume that G(S) is connected and that the smallest non-zero eigenvalue of the Laplace operator ∆µ corresponding to the simple random walk on G(S) satisfies λ1(G(S))> 1
2. Then Γ has Property (T).
We prove that the condition λ1(G(S)) > 1
2 cannot be improved, using two examples.
Example 4. Consider S = { 1,−1,2,−2} a generating set of the group Z and let G(S) be the finite graph associated to S. Then the graph G(S) is the graph:
1
−1 −2
2
w w
w w
Since the Laplace operator ∆µ is defined by:
(∆µf)(x) =f(x)−X
x∼y
f(y)µ(x, y),
and
µ(x, y) =
1
deg(x) if(x, y)∈S×S
0 otherwise
(0.2)
Then the matrix of the Laplace operator ∆µwith respect to the basis{δs|s∈S}
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Surveys in Mathematics and its Applications6(2011), 161 – 164 http://www.utgjiu.ro/math/sma
A note of Zuk’s criterion 163 is the following matrix:
A=
1 −1 2 −1
2 0
−1 1 0 0
−1
2 0 1 −1
2
0 0 −1 1
(0.3)
Then det(A−αI4) = (1−α)2[(1−α)2−3 4]−1
2(1−α)2+1 4 = 0
⇒α∈ {0,1 2,3
2,2} ⇒λ1(G(S)) = 1 2.
But Zdoes not have Property (T).( see [1])
Example 5. The group SL2(Z) is generated by the matrices A =
1 1 0 1
! and
B =
0 −1
1 0
! .
We consider the following generating set of the group SL2(Z):
S={−I, A, B,−A,−B, A−1, B−1,−A−1,−B−1} . The graph G(S) is:
J
J J
J J
J J Z
Z Z
Z Z
Z Z
Z ZZ HH
HH HH
HH HH
HH HH PP
PP PP
PP PP
PP PP
PP PP
PPP t
t t
t t
t t
t t
−I
−A −A−1 −B −B−1
A A−1 B B−1
Then the matrix of Laplace operator∆µ with respect to the basis {δs|s∈S} is the following matrix:
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Surveys in Mathematics and its Applications6(2011), 161 – 164 http://www.utgjiu.ro/math/sma
164 T. Preda
A=
1 1
8 1 8
1 8
1 8
1 8
1 8
1 8
1 8 1
2 1 1
2 0 0 0 0 0 0
1 2
1
2 1 0 0 0 0 0 0
1
2 0 0 1 1
2 0 0 0 0
1
2 0 0 1
2 1 0 0 0 0
1
2 0 0 0 0 1 1
2 0 0
1
2 0 0 0 0 1
2 1 0 0
1
2 0 0 0 0 0 0 1 1
2 1
2 0 0 0 0 0 0 1
2 1
(0.4)
Computing det(A−αI9) = [(1−α)2−1 4]3(3
2−α)(α2−5
2α) = 0⇒
⇒α∈ {0,1 2,3
2,5
2} ⇒λ1(G(S)) = 1 2.
But SL2(Z) does not have Property (T). (see [1]) These two examples shows that 1
2 is the best constant in Zuk’s criterion and cannot be improved.
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, Bucure¸sti, Romania.
e-mail: [email protected]
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Surveys in Mathematics and its Applications6(2011), 161 – 164 http://www.utgjiu.ro/math/sma
