×

Expansion in \(\mathrm{SL}_d(\mathcal O_K/I)\), \(I\) square-free. (English) Zbl 1269.20044

Let \(S\) be a fixed symmetric finite subset of \(\mathrm{SL}_d(\mathcal O_K)\) that generates a Zariski dense subgroup of \(\mathrm{SL}_d(\mathcal O_K)\) when considered as an algebraic group over \(\mathbb Q\) by restriction of scalars. The author proves that the Cayley graphs of \(\mathrm{SL}_d(\mathcal O_K/I)\) with respect to the projections of \(S\) is an expander family when \(I\) ranges over square-free ideals of \(\mathcal O_K\) if \(d=2\) and \(K\) is an arbitrary number field or if \(d=3\) and \(K=\mathbb Q\).

MSC:

20G30 Linear algebraic groups over global fields and their integers
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11B30 Arithmetic combinatorics; higher degree uniformity
11B75 Other combinatorial number theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abels, H., Margulis, G. A., Soifer, G. A.: Semigroups containing proximal linear maps. Israel J. Math. 91, 1-30 (1995) · Zbl 0845.22004 · doi:10.1007/BF02761637
[2] Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83-96 (1986) · Zbl 0661.05053 · doi:10.1007/BF02579166
[3] Alon, N., D. Milman, V.: \lambda 1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38, 73-88 (1985) · Zbl 0549.05051 · doi:10.1016/0095-8956(85)90092-9
[4] Berenstein, C. A., Yger, A.: Effective Bezout identities in Q[z1, . . . , zn]. Acta Math. 166, 69-120 (1991) · Zbl 0724.32002 · doi:10.1007/BF02398884
[5] Bourgain, J., Gamburd, A.: Uniform expansion bounds for Cayley graphs of SL2(Fp), Ann. of Math. 167, 625-642 (2008) · Zbl 1216.20042 · doi:10.4007/annals.2008.167.625
[6] Bourgain, J., Gamburd, A.: Expansion and random walks in SLd (Z/pnZ): I. J. Eur. Math. Soc. 10, 987-1011 (2008) · Zbl 1193.20059 · doi:10.4171/JEMS/137
[7] Bourgain, J., Gamburd, A.: Expansion and random walks in SLd (Z/pnZ): II. With an · Zbl 1193.20060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.