By Mal'tsev N., Kuz'min E. N.
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Additional info for A basis for the identities of the algebra of second-order matrices over a finite field
2 Cayley Graphs Suppose X is a set of generators for a group G, closed under the taking of inverses. The Cayley graph of (G, X) is the graph G(G, E), where E is the subset of G × G deﬁned by the condition that (x, y) ∈ E if and only if xy −1 ∈ X (or equivalently yx−1 ∈ X). The group G acts simply transitively and isometrically on G(G, E) by left multiplication, so that Cayley graphs are homogeneous: all points “look alike”. Cayley graphs are good for obtaining examples of graphs of small degree and small diameter but high cardinality (these are “expanders”, which are important in discrete mathematics).
The representation π1 is a multiple of the trivial representation, and the associated matrix coeﬃcients are constants, while all the matrix coeﬃcients of π0 vanish at inﬁnity in G. Several of the proofs of this involve looking at subgroups R of G similar to the group Q described above, and “lifting” to G the decomposition from R. The diﬃculty of the proof is in showing that G acts trivially on the vectors where the normal subgroup N of R acts trivially. The remarkable fact is that we can often say more than this: for most representations of interest, there is control on the rate of decay.
M. Rosenblatt, D. A. Margulis [82, 88, 98]) that Lebesgue measure is the only ﬁnitely additive rotation-invariant position additive set function on the sphere S k , for k ≥ 5, and the construction by Margulis [81, 83] of “expanders”, graphs with a very high degree of connectivity. The monograph of P. de la Harpe and A. Valette  presents a detailed account of these applications, and much more; for more recent applications, see also the monographs of P. Sarnak  and A. Lubotzky . If G is a simple Lie group with property T , then there exists pG in (2, ∞) such that π(·)ξ, η ∈ LpG + (G) ∀ξ, η ∈ Hπ for all unitary representations π of G with no trivial subrepresentations, and further ∀ξ, η ∈ Hπ .
A basis for the identities of the algebra of second-order matrices over a finite field by Mal'tsev N., Kuz'min E. N.