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Universal lattices and unbounded rank expanders. (English) Zbl 1140.20039

It is conjectured that the groups \(\text{SL}_d(\mathbb{Z}[x_1,\dots,x_k])\), called universal lattices, have property \(T\). In this paper it are studied noncommutative analogs of these groups, i.e., the subgroups of \(\text{GL}_d(\mathbb{Z}\langle x_1,\dots,x_k\rangle)\) generated by all elementary matrices. As these groups can be mapped onto many universal lattices, one can obtain better bounds for the \(\tau\)-constant for the groups \(\text{SL}_d(\mathbb{Z}[x_1,\dots,x_k])\).
Let \(R\) be an associative ring with unit, and let \(E_{i,j}\) denote the set of elementary \(d\times d\) matrices \(\{\text{Id}+r\cdot e_{i,j}\mid r\in R\}\). Set \(E=E(R)=\bigcup_{i\neq j}E_{i,j}\) and let \(\text{EL}_d(R)\) be the subgroup of the multiplicative group of the ring of \(d\times d\) matrices over \(R\) generated by \(E(R)\). The group \(G=\text{EL}_d(R)\) is said to have bounded elementary generation if there is a number \(N=BE_d(R)\) such that every element of \(G\) can be written as a product of at most \(N\) elements from the set \(E\).
The author proves that if \(d\geq 3\) and \(R\) is a finitely generated associative ring such that \(\text{EL}_d(R)\) has bounded elementary generation, then \(\text{EL}_d(R)\) has property \(T\). Moreover, there is an explicit lower bound for the Kazhdan constant. It is proved that the commutative universal lattices \(\text{SL}_d(\mathbb{Z}[x_1,\dots,x_k])\) have property \(\tau\), for \(d\geq 3\), and the \(\tau\)-constant with respect to the generating set consisting of all elementary matrices with \(\pm 1\) off the diagonal or with \(\pm x_i\) next to the main diagonal, is bounded from below by \(1/(800\sqrt d(1+(k/d)^{3/2}))\).
For all primes \(p\) and integers \(l\), there exists a generating set \(\Sigma'_{3l}\) with 28 elements of the group \(\text{SL}_{3l}(F_p)\) such that the Kazhdan constant \({\mathcal K}(\text{SL}_{3l}(F_p);\Sigma'_{3l})>1/400\) for all \(l,p\). Therefore, the corresponding Cayley graphs form a family of expanders with expanding constant at least \(10^{-6}\). – This provides the first example of expander families of groups of Lie type, where the rank is not bounded and provides counter examples to two conjectures of A. Lubotzky and B. Weiss.

MSC:

20H25 Other matrix groups over rings
20F05 Generators, relations, and presentations of groups
20G35 Linear algebraic groups over adèles and other rings and schemes
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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