Uniform Kazhdan Constant for some families of linear groups
نویسنده
چکیده
Let R be a ring generated by l elements with stable range r. Assume that the group ELd(R) has Kazhdan constant ǫ0 > 0 for some d ≥ r + 1. We prove that there exist ǫ(ǫ0, l) > 0 and k ∈ N, s.t. for every n ≥ d, ELn(R) has a generating set of order k and a Kazhdan constant larger than ǫ. As a consequence, we obtain for SLn(Z) where n ≥ 3, a Kazhdan constant which is independent of n w.r.t generating set of a fixed size.
منابع مشابه
Symmetric Groups and Expanders
We construct an explicit generating sets Fn and F̃n of the alternating and the symmetric groups, which make the Cayley graphs C(Alt(n), Fn) and C(Sym(n), F̃n) a family of bounded degree expanders for all sufficiently large n. These expanders have many applications in the theory of random walks on groups and other areas of mathematics. A finite graph Γ is called an ǫ-expander for some ǫ ∈ (0, 1), ...
متن کاملLinear Algebra Construction of Formal Kazhdan-lusztig Bases
General facts of linear algebra are used to give proofs for the (wellknown) existence of analogs of Kazhdan-Lusztig polynomials corresponding to formal analogs of the Kazhdan-Lusztig involution, and of explicit formulae (some new, some known) for their coefficients in terms of coefficients of other natural families of polynomials (such as the corresponding formal analogs of the Kazhdan-Lusztig ...
متن کاملUniform Kazhdan Constant for {SLn(Z)}n≥3
Let R be a ring generated by l elements with stable range r. Assume that the group EL d (R) has Kazhdan constant ǫ 0 > 0 for some d ≥ r + 1. We prove that there exist ǫ(ǫ 0 , l) > 0 and k ∈ N, s.t. for every n ≥ d, EL n (R) has a generating set of order k and a Kazhdan constant larger than ǫ. As a consequence, we obtain for SL n (Z) where n ≥ 3, a Kazhdan constant which is independent of n w.r....
متن کاملUniform Kazhdan
We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.
متن کاملKazhdan Constants for Sln(Z)
In this article we improve the known Kazhdan constant for SLn(Z) with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by [42 √ n + 860], which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since √ 2/n is an upper bound. We can use this bound to improve the bounds for the spectral gap of the ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008