By Terence Tao
Expander graphs are an incredible device in theoretical desktop technological know-how, geometric crew conception, chance, and quantity idea. moreover, the options used to carefully determine the growth estate of a graph draw from such varied components of arithmetic as illustration thought, algebraic geometry, and mathematics combinatorics. this article specializes in the latter subject within the vital case of Cayley graphs on finite teams of Lie sort, constructing instruments similar to Kazhdan's estate (T), quasirandomness, product estimates, break out from subvarieties, and the Balog-Szemeredi-Gowers lemma. functions to the affine sieve of Bourgain, Gamburd, and Sarnak also are given. the cloth is basically self-contained, with extra sections at the normal conception of expanders, spectral idea, Lie concept, and the Lang-Weil sure, in addition to quite a few workouts and different not obligatory material.
About the Author
Terence Tao , college of California, la, CA, united states.
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Extra resources for Expansion in Finite Simple Groups of Lie Type (Graduate Studies in Mathematics)
Infinite, finitely generated groups G with property (i) or (ii) of the above exercise are known as amenable groups; amenability is an important property in ergodic theory, operator algebras, and many other areas of mathematics, but will not be discussed extensively in this course. The notion of amenability can also be extended to other locally compact groups, but we again will not discuss these matters here. From the above exercise, we see that an infinite amenable finitely generated group cannot have property (T).
If (X, µ) is a measure space that G acts on in a transitive measure-preserving fashion, then we have the (left) quasiregular representation τX : G → U (L2 (X, µ)), in which the Hilbert space is L2 (X, µ), and the action is given by the formula τX (g)f (x) := f (g −1 x) for g ∈ G and x ∈ X. Of course, the regular representation can be viewed as a special case of a quasiregular representation, as can the one-dimensional trivial representation. 12 (Direct sum). If ρ1 : G → U (H1 ) and ρ2 : G → U (H2 ) are unitary representations of a locally compact group G, then their direct sum ρ1 ⊕ ρ2 : G → U (H1 ⊕ H2 ) is also a unitary representation, where H1 ⊕ H2 is the Hilbert space of all formal sums v1 ⊕ v2 with v1 ∈ H1 and v2 ∈ H2 with the inner product v1 ⊕ v2 , w1 ⊕ w2 H1 ⊕H2 := v1 , w1 H1 + v2 , w2 H2 and the representation ρ1 ⊕ ρ2 is given by the formula (ρ1 ⊕ ρ2 )(g)(v1 ⊕ v2 ) := (ρ1 (g)v1 ) ⊕ (ρ2 (g)v2 ).
Now we turn to the higher-dimensional cases d ≥ 3. The idea is to first use Fourier analysis17 to understand the action of various simpler subgroups of SLd (R) acting on a space H with approximately invariant vectors, and obtain non-trivial vectors that are invariant with respect to those simpler subgroups. Then, we will use an asymptotic conjugation trick of Mautner to boost this invariance up to increasingly larger groups, until we obtain a non-trivial vector invariant under the whole group SLd (R).